Dual Little Strings and their Partition Functions

Brice Bastian§, Stefan Hohenegger§, Amer Iqbal†⋆,  Soo-Jong Rey
§ Université de Lyon UMR 5822, CNRS/IN2P3, Institut de Physique Nucléaire de Lyon, 4 rue Enrico Fermi, 69622 Villeurbanne Cedex FRANCE
Abdus Salam School of Mathematical Sciences, G.C. University, Lahore PAKISTAN
Center for Theoretical Physics, Lahore, PAKISTAN
School of Physics and Astronomy & Center for Theoretical Physics, Seoul National University, Seoul 08826 KOREA
(March 10, 2024)
Abstract

We study the topological string partition function of a class of toric, double elliptically fibered Calabi-Yau threefolds XN,Msubscript𝑋𝑁𝑀X_{N,M} at a generic point in the Kähler moduli space. These manifolds engineer little string theories in five dimensions or lower and are dual to stacks of M5-branes probing a transverse orbifold singularity. Using the refined topological vertex formalism, we explicitly calculate a generic building block which allows to compute the topological string partition function of XN,Msubscript𝑋𝑁𝑀X_{N,M} as a series expansion in different Kähler parameters. Using this result we give further explicit proof for a duality found previously in the literature, which relates XN,MXN,Msimilar-tosubscript𝑋𝑁𝑀subscript𝑋superscript𝑁superscript𝑀X_{N,M}\sim X_{N^{\prime},M^{\prime}} for NM=NM𝑁𝑀superscript𝑁superscript𝑀NM=N^{\prime}M^{\prime} and gcd(N,M)=gcd(N,M)gcd𝑁𝑀gcdsuperscript𝑁superscript𝑀\text{gcd}(N,M)=\text{gcd}(N^{\prime},M^{\prime}).


I Introduction

Recently, the study of little string theories (LSTs) has received renewed interest: first proposed two decades ago Berkooz:1997cq ; Seiberg:1997zk ; Losev:1997hx ; Aharony:1998ub (see Aharony:1999ks ; Kutasov:2001uf for a review), during the last years, the construction of LSTs from various M-brane constructions and their dual geometric description in F-theory Bhardwaj:2015oru has led to a better understanding of supersymmetric quantum theories in six dimensions. Moreover, in the process new and unexpected dualities have been unravelled. The geometric description of these in terms of F-theory compactification on Calabi-Yau threefolds has been particularly useful in understanding various stringy properties such as LST T-duality Seiberg:1997zk ; Intriligator:1997dh ; Intriligator:1999cn .

A particular class of LSTs, dual to N𝑁N M5-branes transverse to a Msubscript𝑀\mathbb{Z}_{M} orbifold, preserving eight supercharges can be realized in F-theory using toric, non-compact Calabi-Yau manifolds denoted by XN,Msubscript𝑋𝑁𝑀X_{N,M} Haghighat:2013gba ; Hohenegger:2015btj . The latter have the structure of a double elliptic fibration, in which one elliptic fibration has a singularity of type IN1subscript𝐼𝑁1I_{N-1} and the other one of IM1subscript𝐼𝑀1I_{M-1}. The exchange of the two elliptic fibrations is related to the T-duality of LST Bhardwaj:2015oru ; Hohenegger:2015btj . Moreover, using for example the refined topological vertex formalism, the topological string partition function 𝒵N,Msubscript𝒵𝑁𝑀\mathcal{Z}_{N,M} on XN,Msubscript𝑋𝑁𝑀X_{N,M} can be computed explicitly Hohenegger:2015btj . By analysing the web diagrams associated with XN,Msubscript𝑋𝑁𝑀X_{N,M}, it was shown in Hohenegger:2016yuv that XN,MXN,Msimilar-tosubscript𝑋𝑁𝑀subscript𝑋superscript𝑁superscript𝑀X_{N,M}\sim X_{N^{\prime},M^{\prime}} for MN=MN𝑀𝑁superscript𝑀superscript𝑁MN=M^{\prime}N^{\prime} and gcd(M,N)=k=gcd(M,N)gcd𝑀𝑁𝑘gcdsuperscript𝑀superscript𝑁\text{gcd}(M,N)=k=\text{gcd}(M^{\prime},N^{\prime}), i.e. the two Calabi-Yau threefolds lie in the same extended Kähler moduli space and can be related by flop transitions. The partition function of the two dual theories should be the same and so it was expected that the topological string partition function of XN,Msubscript𝑋𝑁𝑀X_{N,M} and XN,Msubscript𝑋superscript𝑁superscript𝑀X_{N^{\prime},M^{\prime}} should be the same i.e.

𝒵N,M(ω,ϵ1,2)=𝒵N,M(ω,ϵ1,2)subscript𝒵𝑁𝑀𝜔subscriptitalic-ϵ12subscript𝒵superscript𝑁superscript𝑀superscript𝜔subscriptitalic-ϵ12\displaystyle\displaystyle\mathcal{Z}_{N,M}(\omega,\epsilon_{1,2})=\mathcal{Z}_{N^{\prime},M^{\prime}}(\omega^{\prime},\epsilon_{1,2})\, (I.1)

for ω𝜔\omega and ωsuperscript𝜔\omega^{\prime} the Kähler forms of the two Calabi-Yau threefolds related by flop transitions. Using a general building block W{β}{α}subscriptsuperscript𝑊𝛼𝛽W^{\{\alpha\}}_{\{\beta\}} to compute 𝒵N,M(ω,ϵ1,2)subscript𝒵𝑁𝑀𝜔subscriptitalic-ϵ12\mathcal{Z}_{N,M}(\omega,\epsilon_{1,2}), we explicitly confirm Eq.(I.1) for gcd(N,M)=1gcd𝑁𝑀1\text{gcd}(N,M)=1, thus verifying the duality proposed in Hohenegger:2016yuv explicitly at the level of the partition function.

This paper is organised as follows. In section 2, we discuss the (N,M)=(3,2)𝑁𝑀32(N,M)=(3,2) case in detail and show that it is dual to the (6,1)61(6,1) case, i.e. related by a combination of flop and symmetry transforms. We also discuss the relation between the general (N,M)𝑁𝑀(N,M) and (k,MN/k)𝑘𝑀𝑁𝑘(k,MN/k) case. In section 3, we calculate the partition function of the (3,2)32(3,2) case and show that, after a change of Kähler parameters, it is equal to the partition function of the (6,1)61(6,1) configuration. In section 4, we discuss the open string amplitudes which make up the building block for generic brane configurations. We also discuss the general “twisted” (1,L)1𝐿(1,L) case, which is related by flop transitions to the standard (1,L)1𝐿(1,L) case. In section 5, we present our conclusions and future directions. Some of the detailed calculations and our notation and conventions are relegated to three appendices.

II Little Strings and Dual Brane Webs

At low energies, a stack of N𝑁N coincident M5-branes engineers a six-dimensional 𝒩=(2,0)𝒩20{\cal N}=(2,0) superconformal field theory of AN1subscript𝐴𝑁1A_{N-1}-type. Upon replacing the 5superscript5\mathbb{R}^{5} transverse to the M5-branes by an trans×4/Msubscripttranssuperscript4subscript𝑀\mathbb{R}_{\text{trans}}\times\mathbb{R}^{4}/\mathbb{Z}_{M} orbifold geometry, the world-volume theory on the branes becomes an 𝒩=(1,0)𝒩10{\cal N}=(1,0) SCFT. One may move away from the conformal point by separating the M5-branes along the transverse transsubscripttrans\mathbb{R}_{\text{trans}}, as massive states appear in the form of M2-branes that end on the separated M5-branes. Note that if transsubscripttrans\mathbb{R}_{\text{trans}} is compactified to a circle 𝕊ρ1subscriptsuperscript𝕊1𝜌\mathbb{S}^{1}_{\rho} of circumference ρ𝜌\rho so that the M5-branes are points on 𝕊ρ1subscriptsuperscript𝕊1𝜌\mathbb{S}^{1}_{\rho}, then the theory arising on the M5-branes (whether coincident or not) gives rise to a LST whose defining scale is ρ𝜌\rho.

As a next step, we compactify the worldvolume of the M5-branes on a circle 𝕊τ1subscriptsuperscript𝕊1𝜏\mathbb{S}^{1}_{\tau} of circumference τ𝜏\tau. This M-brane configuration is dual Cecotti:2013mba to a (p,q)𝑝𝑞(p,q) 5-brane web in type IIB string theory given by N𝑁N coincident NS5-branes wrapped on 𝕊τ1subscriptsuperscript𝕊1𝜏\mathbb{S}^{1}_{\tau} that intersect with M𝑀M coincident D5-branes wrapped on 𝕊ρ1subscriptsuperscript𝕊1𝜌\mathbb{S}^{1}_{\rho}. Turning on deformations which separate the intersections of NS5-branes and D5-branes and stretching the NS5-D5 bound-state branes Aharony:1997bh yields the brane web shown in Fig. 1. Finally, this brane web in type IIB string theory is dual to a toric Calabi-Yau threefold Leung:1997tw , which was first studied in Hohenegger:2015btj and was denoted by XN,Msubscript𝑋𝑁𝑀X_{N,M}, and provides a more geometric description of the theory via F-theory compactification Morrison:1996na ; Morrison:1996pp . A general classification of toric and non-toric base manifolds in F-theory models was discussed in Morrison:2012np ; Morrison:2012js and, more recently, in Choi:2017vtd . For general properties of topological strings on elliptic Calabi-Yau threefolds, see Alim:2012ss ; Klemm:2012sx ; Huang:2015sta .

\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots==111==222==333==M𝑀M==111==222==333==M𝑀M-111-222-333-444-555-N𝑁N-111-222-333-444-555-N𝑁N
Figure 1: The 5-brane web in type IIB string theory dual to the Calabi-Yau threefold XN,Msubscript𝑋𝑁𝑀X_{N,M}. The Kähler parameters associated with the horizontal, vertical and diagonal lines in the web are collectively denoted by 𝐡,𝐯𝐡𝐯{\bf h},{\bf v} and 𝐦𝐦{\bf m}, respectively. More precise labelings will be given for concrete examples below (see e.g. Fig. 2(a)).

The brane web shown in Fig. 1 has 3NM3𝑁𝑀3NM parameters, related to the sizes of various line segments corresponding to the 5-branes. Specifically, we denote the parameters associated with horizontal (green) lines as 𝐡={h1,,hNM}𝐡subscript1subscript𝑁𝑀\mathbf{h}=\{h_{1},\ldots,h_{NM}\}, with vertical (red) lines as 𝐯={v1,,vNM}𝐯subscript𝑣1subscript𝑣𝑁𝑀\mathbf{v}=\{v_{1},\ldots,v_{NM}\} and with diagonal (blue) lines as 𝐦={m1,,mNM}𝐦subscript𝑚1subscript𝑚𝑁𝑀\mathbf{m}=\{m_{1},\ldots,m_{NM}\}. Out of these, only NM+2𝑁𝑀2NM+2 are independent (after imposing a number of consistency conditions related to the the periodic identification of the web diagram Haghighat:2013tka ). In the dual Calabi-Yau threefold XN,Msubscript𝑋𝑁𝑀X_{N,M}, these (NM+2)𝑁𝑀2(NM+2) parameters are independent Kähler moduli. For a discussion of the geometry of XN,Msubscript𝑋𝑁𝑀X_{N,M} and the corresponding mirror curves, see Kanazawa:2016tnt .

The supersymmetric or BPS-state counting partition function of the above mentioned LST can be computed as the topological string partition function 𝒵N,M(𝐡,𝐯,𝐦,ϵ1,ϵ2)subscript𝒵𝑁𝑀𝐡𝐯𝐦subscriptitalic-ϵ1subscriptitalic-ϵ2\mathcal{Z}_{N,M}(\mathbf{h},\mathbf{v},\mathbf{m},\epsilon_{1},\epsilon_{2}) of XN,Msubscript𝑋𝑁𝑀X_{N,M}. The latter depends on the Kähler parameters of XN,Msubscript𝑋𝑁𝑀X_{N,M} (taking into account that only NM+2𝑁𝑀2NM+2 of them are independent) as well as two regularisation parameters ϵ1,2subscriptitalic-ϵ12\epsilon_{1,2} introduced to render the partition function well-defined. From the perspective of the field theory that describes the low-energy limit of the LST engineered by XN,Msubscript𝑋𝑁𝑀X_{N,M}, these parameters can be interpreted geometrically as the ΩΩ\Omega-background Nekrasov:2002qd ; Losev:2003py . The quantization of the Kähler parameters in units of unrefined ϵitalic-ϵ\epsilon, which correspond to Coulomb branch parameters Nekrasov:2003rj , leads to a description of the partition function in terms of an irreducible representation of an affine group. For details, see Bastian:2017jje .

Given the web diagram, 𝒵N,Msubscript𝒵𝑁𝑀\mathcal{Z}_{N,M} can be computed efficiently using the (refined) topological vertex formalism: to this end, a preferred direction in the web needs to be chosen, which from the perspective of LSTs determines the parameters that play the role of coupling constants. Concretely, different choices of the preferred direction lead to different (but equivalent) power series expansions of 𝒵N,Msubscript𝒵𝑁𝑀\mathcal{Z}_{N,M} in terms of either 𝐡𝐡\mathbf{h}, 𝐯𝐯\mathbf{v} or 𝐦𝐦\mathbf{m}. In this paper, we shall use this fact to prove a duality (that was first proposed in Hohenegger:2016yuv ) at the level of the partition function for generic values of the Kähler parameters: indeed, it was argued in Hohenegger:2016yuv that XN,Msubscript𝑋𝑁𝑀X_{N,M} and XN,Msubscript𝑋superscript𝑁superscript𝑀X_{N^{\prime},M^{\prime}} are related to each other through a series of SL(2,)𝑆𝐿2SL(2,\mathbb{Z}) symmetry and flop transformations if NM=NM𝑁𝑀superscript𝑁superscript𝑀NM=N^{\prime}M^{\prime} and gcd(N,M)=k=gcd(N,M)gcd𝑁𝑀𝑘gcdsuperscript𝑁superscript𝑀\text{gcd}(N,M)=k=\text{gcd}(N^{\prime},M^{\prime}). This suggests that

𝒵N,M(𝐡,𝐯,𝐦,ϵ1,ϵ2)=𝒵N,M(𝐡,𝐯,𝐦,ϵ1,ϵ2),subscript𝒵𝑁𝑀𝐡𝐯𝐦subscriptitalic-ϵ1subscriptitalic-ϵ2subscript𝒵superscript𝑁superscript𝑀superscript𝐡superscript𝐯superscript𝐦subscriptitalic-ϵ1subscriptitalic-ϵ2\displaystyle\mathcal{Z}_{N,M}(\mathbf{h},\mathbf{v},\mathbf{m},\epsilon_{1},\epsilon_{2})=\mathcal{Z}_{N^{\prime},M^{\prime}}(\mathbf{h}^{\prime},\mathbf{v}^{\prime},\mathbf{m}^{\prime},\epsilon_{1},\epsilon_{2})\,, (II.1)

where the duality map (𝐡,𝐯,𝐦)(𝐡,𝐯,𝐦)𝐡𝐯𝐦superscript𝐡superscript𝐯superscript𝐦(\mathbf{h},\mathbf{v},\mathbf{m})\longmapsto(\mathbf{h}^{\prime},\mathbf{v}^{\prime},\mathbf{m}^{\prime}) was proposed in Hohenegger:2016yuv (see also appendix C for an example), taking into account the consistency conditions on both sides. The relation (II.1) was tested in Hohenegger:2016yuv (based on a conjecture put forward in Hohenegger:2016eqy ) at a particular region in the Kähler moduli space of XN,Msubscript𝑋𝑁𝑀X_{N,M} (with hi=hjsubscript𝑖subscript𝑗h_{i}=h_{j}, vi=vjsubscript𝑣𝑖subscript𝑣𝑗v_{i}=v_{j} and mi=mjsubscript𝑚𝑖subscript𝑚𝑗m_{i}=m_{j} for i,j=1,,NMformulae-sequence𝑖𝑗1𝑁𝑀i,j=1,\ldots,NM) and the Nekrasov-Shatashvili limit ϵ20subscriptitalic-ϵ20\epsilon_{2}\to 0 Nekrasov:2009rc ; Mironov:2009uv . In the following, by computing 𝒵N,Msubscript𝒵𝑁𝑀\mathcal{Z}_{N,M} at a generic point in the Kähler moduli space, we give explicit evidence of (II.1) in general. However, for simplicity, we shall limit ourselves to gcd(N,M)=1gcd𝑁𝑀1\text{gcd}(N,M)=1.

III Partition Function of (3,2)32(3,2) Case

As checking (II.1) is rather complicated for generic (N,M)𝑁𝑀(N,M), before presenting the generic case in the following section, we first discuss as a (nontrivial) example the case (N,M)=(3,2)𝑁𝑀32(N,M)=(3,2). Specifically, we shall demonstrate in the following that

𝒵3,2(ω,ϵ1,ϵ2)=𝒵6,1(ω,ϵ1,ϵ2),subscript𝒵32𝜔subscriptitalic-ϵ1subscriptitalic-ϵ2subscript𝒵61superscript𝜔subscriptitalic-ϵ1subscriptitalic-ϵ2\displaystyle\mathcal{Z}_{3,2}(\omega,\epsilon_{1},\epsilon_{2})=\mathcal{Z}_{6,1}(\omega^{\prime},\epsilon_{1},\epsilon_{2})\,, (III.1)

where ω𝜔\omega and ωsuperscript𝜔\omega^{\prime} denote the (independent) Kähler parameters of XN,Msubscript𝑋𝑁𝑀X_{N,M} and XN,Msubscript𝑋superscript𝑁superscript𝑀X_{N^{\prime},M^{\prime}}, respectively. We shall verify (III.1) explicitly at a generic point in the moduli space of Kähler parameters by performing a novel expansion of the left hand side of this relation.

III.1 Consistency Conditions and Parametrisation

The web diagram of X3,2subscript𝑋32X_{3,2} is shown in Fig. 2(a) and the Kähler parameters labelling various rational curves are given by

h1,,6,subscript16\displaystyle h_{1,\ldots,6}\,, v1,,6,subscript𝑣16\displaystyle v_{1,\ldots,6}\,, m1,,6.subscript𝑚16\displaystyle m_{1,\ldots,6}\,. (III.2)
==111==222==111==222-111-222-333-111-222-333m1subscript𝑚1m_{1}m2subscript𝑚2m_{2}m3subscript𝑚3m_{3}m4subscript𝑚4m_{4}m5subscript𝑚5m_{5}m6subscript𝑚6m_{6}h5subscript5h_{5}h3subscript3h_{3}h1subscript1h_{1}h2subscript2h_{2}h6subscript6h_{6}h4subscript4h_{4}v1subscript𝑣1v_{1}v5subscript𝑣5v_{5}v3subscript𝑣3v_{3}v4subscript𝑣4v_{4}v2subscript𝑣2v_{2}v6subscript𝑣6v_{6}IVIIVIIVIII111444222555333666777101010888111111999121212(a)
\cdots\cdots\cdots\cdots123123123456456456789789789101112101112101112123123123456456456789789789101112101112101112123123123456456456789789789101112101112101112(b)
Figure 2: (a) Web diagram of X3,2subscript𝑋32X_{3,2} with explicit labelling of all line segments. The Roman numerals I,\ldots,VI denote the different hexagons for which the consistency conditions (III.3) – (III.8) are imposed. (b) Newton polygon and tiling of the plane: The grey area is dual to the web diagram of XN,Msubscript𝑋𝑁𝑀X_{N,M}. Due to the periodic identification of the web, the Newton polygon has to be periodically continued, leading to a tiling of the plane. For better readability, we have labelled the vertices of the web-diagram, as well as all of the equivalent dual faces of the Newton polygon.

Of these 18 parameters, however, only 8 are independent: indeed, for each of the six compact divisors of X3,2subscript𝑋32X_{3,2} (each of which is a 1×1superscript1superscript1\mathbb{P}^{1}\times\mathbb{P}^{1} blown up at two points), we have two consistency conditions. These conditions arise because each compact divisor (represented as a hexagon in the web) has six rational curves which are toric divisors of 1×1superscript1superscript1\mathbb{P}^{1}\times\mathbb{P}^{1} blown up at two points but only four of these are independent. Explicitly, we therefore obtain the following conditions:

hexagon I:\displaystyle\bullet\,\text{hexagon I}:
h4+m4=m3+h1,v3+m3=m4+v1,formulae-sequencesubscript4subscript𝑚4subscript𝑚3subscript1subscript𝑣3subscript𝑚3subscript𝑚4subscript𝑣1\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ h_{4}+m_{4}=m_{3}+h_{1}\,,v_{3}+m_{3}=m_{4}+v_{1}\,, (III.3)
hexagon II:\displaystyle\bullet\,\text{hexagon II}:
h5+m2=m4+h2,v4+m4=m2+v2,formulae-sequencesubscript5subscript𝑚2subscript𝑚4subscript2subscript𝑣4subscript𝑚4subscript𝑚2subscript𝑣2\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ h_{5}+m_{2}=m_{4}+h_{2}\,,v_{4}+m_{4}=m_{2}+v_{2}\,, (III.4)
hexagon III:\displaystyle\bullet\,\text{hexagon III}:
h6+m6=m2+h3,v5+m2=m6+v3,formulae-sequencesubscript6subscript𝑚6subscript𝑚2subscript3subscript𝑣5subscript𝑚2subscript𝑚6subscript𝑣3\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ h_{6}+m_{6}=m_{2}+h_{3}\,,v_{5}+m_{2}=m_{6}+v_{3}\,, (III.5)
hexagon IV:\displaystyle\bullet\,\text{hexagon IV}:
h1+m1=m6+h4,v6+m6=m1+v4,formulae-sequencesubscript1subscript𝑚1subscript𝑚6subscript4subscript𝑣6subscript𝑚6subscript𝑚1subscript𝑣4\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ h_{1}+m_{1}=m_{6}+h_{4}\,,v_{6}+m_{6}=m_{1}+v_{4}\,, (III.6)
hexagon V:\displaystyle\bullet\,\text{hexagon V}:
h2+m5=m1+h5,v1+m1=m5+v5,formulae-sequencesubscript2subscript𝑚5subscript𝑚1subscript5subscript𝑣1subscript𝑚1subscript𝑚5subscript𝑣5\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ h_{2}+m_{5}=m_{1}+h_{5}\,,v_{1}+m_{1}=m_{5}+v_{5}\,, (III.7)
hexagon VI:\displaystyle\bullet\,\text{hexagon VI}:
h3+m3=m5+h6,v2+m5=m3+v6.formulae-sequencesubscript3subscript𝑚3subscript𝑚5subscript6subscript𝑣2subscript𝑚5subscript𝑚3subscript𝑣6\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ h_{3}+m_{3}=m_{5}+h_{6}\,,v_{2}+m_{5}=m_{3}+v_{6}\,. (III.8)

The above conditions indeed leave 888 independent parameters. However, there is a priori an ambiguity in choosing the latter. A suitable parametrisation can be found by comparison with the web diagram of X6,1subscript𝑋61X_{6,1} in Fig. 3, which has been shown in Hohenegger:2016yuv to be dual to X3,2subscript𝑋32X_{3,2} by flop and symmetry transforms. The (6,1)61(6,1) diagram is parametrised in terms of the maximal set of variables (M,V,H1,,6)𝑀𝑉subscript𝐻16(M,V,H_{1,\ldots,6}), which satisfies all consistency conditions. Through the explicit duality map found in Hohenegger:2016yuv , we can also express the (𝐡,𝐯,𝐦)𝐡𝐯𝐦(\mathbf{h},\mathbf{v},\mathbf{m}) of the (3,2)32(3,2) web diagram in terms of (M,V,H1,,6)𝑀𝑉subscript𝐻16(M,V,H_{1,\ldots,6})

h1=MH5H6,h2=MH1H6formulae-sequencesubscript1𝑀subscript𝐻5subscript𝐻6subscript2𝑀subscript𝐻1subscript𝐻6\displaystyle h_{1}=-M-H_{5}-H_{6}\,,\hskip 8.5359pth_{2}=-M-H_{1}-H_{6}
h3=MH1H2,h4=MH2H3formulae-sequencesubscript3𝑀subscript𝐻1subscript𝐻2subscript4𝑀subscript𝐻2subscript𝐻3\displaystyle h_{3}=-M-H_{1}-H_{2}\,,\hskip 8.5359pth_{4}=-M-H_{2}-H_{3}
h5=MH3H4,h6=MH4H5formulae-sequencesubscript5𝑀subscript𝐻3subscript𝐻4subscript6𝑀subscript𝐻4subscript𝐻5\displaystyle h_{5}=-M-H_{3}-H_{4}\,,\hskip 8.5359pth_{6}=-M-H_{4}-H_{5}
v1=2M+H1+H5+H6,v2=2M+H1+H2+H6,formulae-sequencesubscript𝑣12𝑀subscript𝐻1subscript𝐻5subscript𝐻6subscript𝑣22𝑀subscript𝐻1subscript𝐻2subscript𝐻6\displaystyle v_{1}=2M+H_{1}+H_{5}+H_{6}\,,\hskip 2.84544ptv_{2}=2M+H_{1}+H_{2}+H_{6}\,,
v3=2M+H1+H2+H3,v4=2M+H2+H3+H4,formulae-sequencesubscript𝑣32𝑀subscript𝐻1subscript𝐻2subscript𝐻3subscript𝑣42𝑀subscript𝐻2subscript𝐻3subscript𝐻4\displaystyle v_{3}=2M+H_{1}+H_{2}+H_{3}\,,\hskip 2.84544ptv_{4}=2M+H_{2}+H_{3}+H_{4}\,,
v5=2M+H3+H4+H5,v6=2M+H4+H5+H6,formulae-sequencesubscript𝑣52𝑀subscript𝐻3subscript𝐻4subscript𝐻5subscript𝑣62𝑀subscript𝐻4subscript𝐻5subscript𝐻6\displaystyle v_{5}=2M+H_{3}+H_{4}+H_{5}\,,\hskip 2.84544ptv_{6}=2M+H_{4}+H_{5}+H_{6}\,,
m1=2M+H3+2H4+2H5+H6+V,subscript𝑚12𝑀subscript𝐻32subscript𝐻42subscript𝐻5subscript𝐻6𝑉\displaystyle m_{1}=2M+H_{3}+2H_{4}+2H_{5}+H_{6}+V\,,
m2=2M+H1+2H2+2H3+H4+V,subscript𝑚22𝑀subscript𝐻12subscript𝐻22subscript𝐻3subscript𝐻4𝑉\displaystyle m_{2}=2M+H_{1}+2H_{2}+2H_{3}+H_{4}+V\,,
m3=2M+2H1+H2+H5+2H6+V,subscript𝑚32𝑀2subscript𝐻1subscript𝐻2subscript𝐻52subscript𝐻6𝑉\displaystyle m_{3}=2M+2H_{1}+H_{2}+H_{5}+2H_{6}+V\,,
m4=2M+2H1+2H2+H3+H6+V,subscript𝑚42𝑀2subscript𝐻12subscript𝐻2subscript𝐻3subscript𝐻6𝑉\displaystyle m_{4}=2M+2H_{1}+2H_{2}+H_{3}+H_{6}+V\,,
m5=2M+H1+H4+2H5+2H6+V,subscript𝑚52𝑀subscript𝐻1subscript𝐻42subscript𝐻52subscript𝐻6𝑉\displaystyle m_{5}=2M+H_{1}+H_{4}+2H_{5}+2H_{6}+V\,,
m6=2M+H2+2H3+2H4+H5+V.subscript𝑚62𝑀subscript𝐻22subscript𝐻32subscript𝐻4subscript𝐻5𝑉\displaystyle m_{6}=2M+H_{2}+2H_{3}+2H_{4}+H_{5}+V\,. (III.9)

One can indeed check that (III.9) identically satisfies (III.3) – (III.8). Moreover, as we shall discuss in the following, the parametrisation (III.9) is crucial for showing the identity (III.1). To this end, we first review the topological string partition function of X6,1subscript𝑋61X_{6,1} in the following.

III.2 The Partition Function 𝒵6,1subscript𝒵61\mathcal{Z}_{6,1}

The (6,1)61(6,1) web is shown in Fig. 3 in which the Kähler parameters are also labelled.

M𝑀MM𝑀MM𝑀MM𝑀MM𝑀MM𝑀MH1subscript𝐻1H_{1}H2subscript𝐻2H_{2}H3subscript𝐻3H_{3}H4subscript𝐻4H_{4}H5subscript𝐻5H_{5}H6subscript𝐻6H_{6}H1subscript𝐻1H_{1}V𝑉VV𝑉VV𝑉VV𝑉VV𝑉VV𝑉VV𝑉VV𝑉VV𝑉VV𝑉VV𝑉VV𝑉Va𝑎aa𝑎a111222333444555666111222333444555666
Figure 3: Toric diagram dual to the configuration (3,2)32(3,2). This configuration is obtained Hohenegger:2016yuv by performing a flop transformation on the line segments hisubscript𝑖h_{i} and visubscript𝑣𝑖v_{i} (for i=1,,6𝑖16i=1,\ldots,6) (and a combination of additional symmetry transforms) in the original diagram in the left portion of Fig. 2. Notice that the parameters (M,V,H1,,6)𝑀𝑉subscript𝐻16(M,V,H_{1,\ldots,6}) are the same as in Fig. 2.

Here, the parameters (M,V,H1,,6)𝑀𝑉subscript𝐻16(M,V,H_{1,\ldots,6}) are the same that appear in (III.9) (after applying the duality map proposed in Hohenegger:2016yuv ). Furthermore, the partition function 𝒵6,1subscript𝒵61\mathcal{Z}_{6,1} was explicitly calculated in Haghighat:2013gba ; Haghighat:2013tka ; Hohenegger:2013ala using the (refined) topological vertex Aganagic:2003db ; Iqbal:2007ii . The latter has a preferred direction which can be chosen in different ways to obtain different series expansions of 𝒵6,1subscript𝒵61\mathcal{Z}_{6,1}. For example, choosing the preferred direction to be vertical (i.e. along the (0,1)01(0,1)-direction in the web diagram in Fig. 3), the topological string partition function takes the form

𝒵6,1(V,M,Hi,ϵ1,2)=W6()subscript𝒵61𝑉𝑀subscript𝐻𝑖subscriptitalic-ϵ12subscript𝑊6\displaystyle\mathcal{Z}_{6,1}(V,M,H_{i},\epsilon_{1,2})=W_{6}(\emptyset) (III.10)
×α1,,α6(QVQM)|α1|++|α6|(a=16ϑαaαa(QM;ρ)ϑαaαa(t/q;ρ))\displaystyle\times\sum_{\alpha_{1},\ldots,\alpha_{6}}(-Q_{V}Q_{M})^{|\alpha_{1}|+\ldots+|\alpha_{6}|}\left(\prod_{a=1}^{6}\frac{\vartheta_{\alpha_{a}\alpha_{a}}(Q_{M};\rho)}{\vartheta_{\alpha_{a}\alpha_{a}}(\sqrt{t/q};\rho)}\right)
×(1a<b6ϑαaαb(QabQM1;ρ)ϑαaαb(QabQM;ρ)ϑαaαb(Qabt/q;ρ)ϑαaαb(Qabq/t;ρ)),absentsubscriptproduct1𝑎𝑏6subscriptitalic-ϑsubscript𝛼𝑎subscript𝛼𝑏subscript𝑄𝑎𝑏superscriptsubscript𝑄𝑀1𝜌subscriptitalic-ϑsubscript𝛼𝑎subscript𝛼𝑏subscript𝑄𝑎𝑏subscript𝑄𝑀𝜌subscriptitalic-ϑsubscript𝛼𝑎subscript𝛼𝑏subscript𝑄𝑎𝑏𝑡𝑞𝜌subscriptitalic-ϑsubscript𝛼𝑎subscript𝛼𝑏subscript𝑄𝑎𝑏𝑞𝑡𝜌\displaystyle\times\left(\prod_{1\leq a<b\leq 6}\frac{\vartheta_{\alpha_{a}\alpha_{b}}(Q_{ab}Q_{M}^{-1};\rho)\,\vartheta_{\alpha_{a}\alpha_{b}}(Q_{ab}Q_{M};\rho)}{\vartheta_{\alpha_{a}\alpha_{b}}(Q_{ab}\sqrt{t/q};\rho)\,\vartheta_{\alpha_{a}\alpha_{b}}(Q_{ab}\sqrt{q/t};\rho)}\right),

where we adopted the following definitions

QV=eV,subscript𝑄𝑉superscript𝑒𝑉\displaystyle Q_{V}=e^{-V}\,, QM=eM,subscript𝑄𝑀superscript𝑒𝑀\displaystyle Q_{M}=e^{-M}\,, QHi=eHi,subscript𝑄subscript𝐻𝑖superscript𝑒subscript𝐻𝑖\displaystyle Q_{H_{i}}=e^{-H_{i}}\,,
q=e2πiϵ1,𝑞superscript𝑒2𝜋𝑖subscriptitalic-ϵ1\displaystyle q=e^{2\pi i\epsilon_{1}}\,, t=e2πiϵ2.𝑡superscript𝑒2𝜋𝑖subscriptitalic-ϵ2\displaystyle t=e^{-2\pi i\epsilon_{2}}\,. (III.11)

and

Qρ=e2πiρ=e6Mi=16Hi,subscript𝑄𝜌superscript𝑒2𝜋𝑖𝜌superscript𝑒6𝑀superscriptsubscript𝑖16subscript𝐻𝑖\displaystyle Q_{\rho}=e^{2\pi i\rho}=e^{-6M-\sum_{i=1}^{6}H_{i}}\,,
Qab=QMbaQHaQHb1,subscript𝑄𝑎𝑏superscriptsubscript𝑄𝑀𝑏𝑎subscript𝑄subscript𝐻𝑎subscript𝑄subscript𝐻𝑏1\displaystyle Q_{ab}=Q_{M}^{b-a}Q_{H_{a}}\dots Q_{H_{b-1}}\,,  1a<b.for-all1𝑎𝑏\displaystyle\forall\,1\leq a<b\,. (III.12)

The summation is over integer partitions α1,,6subscript𝛼16\alpha_{1,\ldots,6} and ϑμνsubscriptitalic-ϑ𝜇𝜈\vartheta_{\mu\nu} is defined in equation (A.3) in appendix A.

III.3 Diagonal Expansion of 𝒵3,2subscript𝒵32\mathcal{Z}_{3,2}

From the viewpoint of independent variables (M,V,H1,,6)𝑀𝑉subscript𝐻16(M,V,H_{1,\ldots,6}), the (vertical) partition function (III.10) is written as a power series expansion in QVsubscript𝑄𝑉Q_{V}, since neither of the ϑitalic-ϑ\vartheta-functions depends on V𝑉V. Thus, in order to match (III.1) order by order in QVsubscript𝑄𝑉Q_{V}, we have to write 𝒵3,2subscript𝒵32\mathcal{Z}_{3,2} in a similar fashion. Upon inspection of (III.9), this can indeed be achieved by choosing the preferred direction of the refined topological vertex to be along (1,1)11(1,1) (i.e. diagonally in the web diagram in Fig. 2(a)). Indeed, in this manner, the partition function is seen to take the form

𝒵3,2(𝐡,𝐯,𝐦,ϵ1,2)=α1,,α6Qm1|α4|Qm2|α2|Qm3|α6|Qm4|α1|Qm5|α5|Qm6|α3|Wα1α2α3α4α5α6α4α5α6α1α2α3(Qhi,Qvi,ϵ1,2),subscript𝒵32𝐡𝐯𝐦subscriptitalic-ϵ12subscriptsubscript𝛼1subscript𝛼6superscriptsubscript𝑄subscript𝑚1subscript𝛼4superscriptsubscript𝑄subscript𝑚2subscript𝛼2superscriptsubscript𝑄subscript𝑚3subscript𝛼6superscriptsubscript𝑄subscript𝑚4subscript𝛼1superscriptsubscript𝑄subscript𝑚5subscript𝛼5superscriptsubscript𝑄subscript𝑚6subscript𝛼3superscriptsubscript𝑊subscript𝛼1subscript𝛼2subscript𝛼3subscript𝛼4subscript𝛼5subscript𝛼6subscript𝛼4subscript𝛼5subscript𝛼6subscript𝛼1subscript𝛼2subscript𝛼3subscript𝑄subscript𝑖subscript𝑄subscript𝑣𝑖subscriptitalic-ϵ12\displaystyle\mathcal{Z}_{3,2}(\mathbf{h},\mathbf{v},\mathbf{m},\epsilon_{1,2})=\sum_{\alpha_{1},\ldots,\alpha_{6}}Q_{m_{1}}^{|\alpha_{4}|}\,Q_{m_{2}}^{|\alpha_{2}|}\,Q_{m_{3}}^{|\alpha_{6}|}\,Q_{m_{4}}^{|\alpha_{1}|}\,Q_{m_{5}}^{|\alpha_{5}|}\,Q_{m_{6}}^{|\alpha_{3}|}\,W_{\alpha_{1}\,\alpha_{2}\,\alpha_{3}\,\alpha_{4}\,\alpha_{5}\,\alpha_{6}}^{\alpha_{4}\,\alpha_{5}\,\alpha_{6}\,\alpha_{1}\,\alpha_{2}\,\alpha_{3}}(Q_{h_{i}},Q_{v_{i}},\epsilon_{1,2})\,, (III.13)

where α1,,6subscript𝛼16\alpha_{1,\ldots,6} are integer partitions. Furthermore, we used the notation (for i=1,,6𝑖16i=1,\ldots,6)

Qhi=ehi,subscript𝑄subscript𝑖superscript𝑒subscript𝑖\displaystyle Q_{h_{i}}=e^{-h_{i}}\,, Qvi=evi,subscript𝑄subscript𝑣𝑖superscript𝑒subscript𝑣𝑖\displaystyle Q_{v_{i}}=e^{-v_{i}}\,, Qmi=emi,subscript𝑄subscript𝑚𝑖superscript𝑒subscript𝑚𝑖\displaystyle Q_{m_{i}}=e^{-m_{i}}\,, (III.14)

together with (III.11). The coefficient Wα1α2α3α4α5α6α4α5α6α1α2α3superscriptsubscript𝑊subscript𝛼1subscript𝛼2subscript𝛼3subscript𝛼4subscript𝛼5subscript𝛼6subscript𝛼4subscript𝛼5subscript𝛼6subscript𝛼1subscript𝛼2subscript𝛼3W_{\alpha_{1}\,\alpha_{2}\,\alpha_{3}\,\alpha_{4}\,\alpha_{5}\,\alpha_{6}}^{\alpha_{4}\,\alpha_{5}\,\alpha_{6}\,\alpha_{1}\,\alpha_{2}\,\alpha_{3}} shall be computed in the following. As neither 𝐡𝐡\mathbf{h} nor 𝐯𝐯\mathbf{v} depend on V𝑉V, (III.13) is also a series expansion in eVsuperscript𝑒𝑉e^{-V}, which can be compared order by order to (III.10) thereby allowing to verify (III.1).

To our knowledge, while the topological string partition function has been studied in recent literature Haghighat:2013gba ; Haghighat:2013tka ; Hohenegger:2015btj ; Hohenegger:2016yuv ; Hohenegger:2016eqy ; Hohenegger:2013ala ; Bastian:2017jje , diagonal expansions of the form (III.13) for 𝒵N,Msubscript𝒵𝑁𝑀\mathcal{Z}_{N,M} have not been studied so far. From geometric perspective, this expansion has a very natural interpretation: as the vertical and horizontal legs of the (3,2)32(3,2) web diagram are pairwise glued together, the corresponding Newton polygon should be considered on a torus, equivalently, a tiling of the plane should be considered with the fundamental domain (the tile) given in Fig. 2(b). The choice of fundamental domain, however, is not unique and different choices lead to different (but equivalent) presentations of the web. An alternative such choice is shown in Fig. 4(a). Indeed, the green region contains all 12 faces exactly once. This new fundamental domain, however, gives rise to an equivalent representations of the web as given in the Fig. 4(b), which we refer to as the “twisted” (6,1)61(6,1) web. After an SL(2,)𝑆𝐿2SL(2,\mathbb{Z}) transformation, this twisted (6,1)61(6,1) web becomes very similar to the (6,1)61(6,1) web in Fig. 3 except that the upper diagonal and lower diagonal legs that are glued together are not aligned (Indeed, it was shown in Hohenegger:2016yuv that, by a combination of symmetry and flop transforms, the twisted (6,1)61(6,1) web can be converted into the standard (6,1)61(6,1) web which is shown in Fig. 3).

\cdots\cdots\cdots\cdots123123456456789789101112101112123123456456789789101112101112123123456456789789101112101112123123456456789789101112101112(a)
444555666111222333111222333444555666a𝑎aa𝑎ah1subscript1h_{1}h2subscript2h_{2}h3subscript3h_{3}h4subscript4h_{4}h5subscript5h_{5}h6subscript6h_{6}h1subscript1h_{1}v1subscript𝑣1v_{1}v2subscript𝑣2v_{2}v3subscript𝑣3v_{3}v4subscript𝑣4v_{4}v5subscript𝑣5v_{5}v6subscript𝑣6v_{6}m1subscript𝑚1m_{1}m5subscript𝑚5m_{5}m3subscript𝑚3m_{3}m4subscript𝑚4m_{4}m2subscript𝑚2m_{2}m6subscript𝑚6m_{6}m4subscript𝑚4m_{4}m2subscript𝑚2m_{2}m6subscript𝑚6m_{6}m1subscript𝑚1m_{1}m5subscript𝑚5m_{5}m3subscript𝑚3m_{3}(b)
Figure 4: (a) Different choice of fundamental polygon in the tiling of the plane for the configuration (N,M)=(3,2)𝑁𝑀32(N,M)=(3,2). (b) The twisted web diagram as the dual of the green-colored fundamental domain.

III.4 Diagonal Partition Function

Using the presentation of (3,2)32(3,2) diagram shown in Fig. 4(b), it remains to compute the coefficient Wα1α2α3α4α5α6α4α5α6α1α2α3superscriptsubscript𝑊subscript𝛼1subscript𝛼2subscript𝛼3subscript𝛼4subscript𝛼5subscript𝛼6subscript𝛼4subscript𝛼5subscript𝛼6subscript𝛼1subscript𝛼2subscript𝛼3W_{\alpha_{1}\,\alpha_{2}\,\alpha_{3}\,\alpha_{4}\,\alpha_{5}\,\alpha_{6}}^{\alpha_{4}\,\alpha_{5}\,\alpha_{6}\,\alpha_{1}\,\alpha_{2}\,\alpha_{3}} in (III.13) to obtain 𝒵3,2subscript𝒵32\mathcal{Z}_{3,2}. To facilitate this computation (and also to explain the structure of the αisubscript𝛼𝑖\alpha_{i} that appear in this expression), we redraw Fig. 4(b) in Fig. 5 to include the integer partitions associated with each interval. Here, we have also indicated the parameter ρ𝜌\rho, which shall play an important role in the explicit computation below.

α4subscript𝛼4\alpha_{4}α5subscript𝛼5\alpha_{5}α6subscript𝛼6\alpha_{6}α1subscript𝛼1\alpha_{1}α2subscript𝛼2\alpha_{2}α3subscript𝛼3\alpha_{3}α1tsubscriptsuperscript𝛼𝑡1\alpha^{t}_{1}α2tsubscriptsuperscript𝛼𝑡2\alpha^{t}_{2}α3tsubscriptsuperscript𝛼𝑡3\alpha^{t}_{3}α4tsubscriptsuperscript𝛼𝑡4\alpha^{t}_{4}α5tsubscriptsuperscript𝛼𝑡5\alpha^{t}_{5}α6tsubscriptsuperscript𝛼𝑡6\alpha^{t}_{6}a𝑎aa𝑎ah1,μ1subscript1subscript𝜇1h_{1},\mu_{1}h2,μ2subscript2subscript𝜇2h_{2},\mu_{2}h3,μ3subscript3subscript𝜇3h_{3},\mu_{3}h4,μ4subscript4subscript𝜇4h_{4},\mu_{4}h5,μ5subscript5subscript𝜇5h_{5},\mu_{5}h6,μ6subscript6subscript𝜇6h_{6},\mu_{6}h1,μ1subscript1subscript𝜇1h_{1},\mu_{1}v1,ν1subscript𝑣1subscript𝜈1v_{1},\nu_{1}v2,ν2subscript𝑣2subscript𝜈2v_{2},\nu_{2}v3,ν3subscript𝑣3subscript𝜈3v_{3},\nu_{3}v4,ν4subscript𝑣4subscript𝜈4v_{4},\nu_{4}v5,ν5subscript𝑣5subscript𝜈5v_{5},\nu_{5}v6,ν6subscript𝑣6subscript𝜈6v_{6},\nu_{6}m1subscript𝑚1m_{1}m5subscript𝑚5m_{5}m3subscript𝑚3m_{3}m4subscript𝑚4m_{4}m2subscript𝑚2m_{2}m6subscript𝑚6m_{6}m4subscript𝑚4m_{4}m2subscript𝑚2m_{2}m6subscript𝑚6m_{6}m1subscript𝑚1m_{1}m5subscript𝑚5m_{5}m3subscript𝑚3m_{3}H5+Msubscript𝐻5𝑀H_{5}+MH6+Msubscript𝐻6𝑀H_{6}+MH1+Msubscript𝐻1𝑀H_{1}+MH2+Msubscript𝐻2𝑀H_{2}+MH3+Msubscript𝐻3𝑀H_{3}+MH4+Msubscript𝐻4𝑀H_{4}+MM𝑀MV2M𝑉2𝑀V-2M
Figure 5: Detailed parametrisation of the twisted web diagram associated with X3,2subscript𝑋32X_{3,2}: The choice of Kähler parameters (M,V,H1,,6)𝑀𝑉subscript𝐻16(M,V,H_{1,\ldots,6}) is inspired by comparison to the web diagram of X6,1subscript𝑋61X_{6,1} shown in Fig. 2. Furthermore, the diagonal intervals are labelled by integer partitions α1,,6subscript𝛼16\alpha_{1,\ldots,6} (and their transposed versions α1,,6tsubscriptsuperscript𝛼𝑡16\alpha^{t}_{1,\ldots,6}), which indicate the gluing of the web.

With this notation, we have the following expression for W𝑊W in terms of the refined topological vertex

Wα1α2α3α4α5α6α4α5α6α1α2α3(Qhi,Qvi,ϵ1,ϵ2)=μ1,,μ6ν1,,ν6(Qh1)|μ1|(Qh2)|μ2|(Qh3)|μ3|(Qh4)|μ4|superscriptsubscript𝑊subscript𝛼1subscript𝛼2subscript𝛼3subscript𝛼4subscript𝛼5subscript𝛼6subscript𝛼4subscript𝛼5subscript𝛼6subscript𝛼1subscript𝛼2subscript𝛼3subscript𝑄subscript𝑖subscript𝑄subscript𝑣𝑖subscriptitalic-ϵ1subscriptitalic-ϵ2subscriptFRACOPsubscript𝜇1subscript𝜇6subscript𝜈1subscript𝜈6superscriptsubscript𝑄subscript1subscript𝜇1superscriptsubscript𝑄subscript2subscript𝜇2superscriptsubscript𝑄subscript3subscript𝜇3superscriptsubscript𝑄subscript4subscript𝜇4\displaystyle W_{\alpha_{1}\,\alpha_{2}\,\alpha_{3}\,\alpha_{4}\,\alpha_{5}\,\alpha_{6}}^{\alpha_{4}\,\alpha_{5}\,\alpha_{6}\,\alpha_{1}\,\alpha_{2}\,\alpha_{3}}(Q_{h_{i}},Q_{v_{i}},\epsilon_{1},\epsilon_{2})=\sum_{{\mu_{1},\ldots,\mu_{6}}\atop{\nu_{1},\ldots,\nu_{6}}}(-Q_{h_{1}})^{|\mu_{1}|}\,(-Q_{h_{2}})^{|\mu_{2}|}\,(-Q_{h_{3}})^{|\mu_{3}|}\,(-Q_{h_{4}})^{|\mu_{4}|}\,
×(Qh5)|μ5|(Qh6)|μ6|(Qv1)|ν1|(Qv2)|ν2|(Qv3)|ν3|(Qv4)|ν4|(Qv5)|ν5|(Qv6)|ν6|absentsuperscriptsubscript𝑄subscript5subscript𝜇5superscriptsubscript𝑄subscript6subscript𝜇6superscriptsubscript𝑄subscript𝑣1subscript𝜈1superscriptsubscript𝑄subscript𝑣2subscript𝜈2superscriptsubscript𝑄subscript𝑣3subscript𝜈3superscriptsubscript𝑄subscript𝑣4subscript𝜈4superscriptsubscript𝑄subscript𝑣5subscript𝜈5superscriptsubscript𝑄subscript𝑣6subscript𝜈6\displaystyle\hskip 14.22636pt\times(-Q_{h_{5}})^{|\mu_{5}|}\,(-Q_{h_{6}})^{|\mu_{6}|}\,(-Q_{v_{1}})^{|\nu_{1}|}\,(-Q_{v_{2}})^{|\nu_{2}|}\,(-Q_{v_{3}})^{|\nu_{3}|}\,(-Q_{v_{4}})^{|\nu_{4}|}\,(-Q_{v_{5}})^{|\nu_{5}|}\,(-Q_{v_{6}})^{|\nu_{6}|}\,
×Cμ1tν1α4(q,t)Cμ2ν1t,α1t(t,q)Cμ2tν2α5(q,t)Cμ3ν2tα2t(t,q)Cμ6tν3α3(q,t)Cμ4ν3tα3t(t,q)absentsubscript𝐶subscriptsuperscript𝜇𝑡1subscript𝜈1subscript𝛼4𝑞𝑡subscript𝐶subscript𝜇2subscriptsuperscript𝜈𝑡1subscriptsuperscript𝛼𝑡1𝑡𝑞subscript𝐶subscriptsuperscript𝜇𝑡2subscript𝜈2subscript𝛼5𝑞𝑡subscript𝐶subscript𝜇3superscriptsubscript𝜈2𝑡subscriptsuperscript𝛼𝑡2𝑡𝑞subscript𝐶subscriptsuperscript𝜇𝑡6subscript𝜈3subscript𝛼3𝑞𝑡subscript𝐶subscript𝜇4subscriptsuperscript𝜈𝑡3subscriptsuperscript𝛼𝑡3𝑡𝑞\displaystyle\hskip 14.22636pt\times C_{\mu^{t}_{1}\nu_{1}\alpha_{4}}(q,t)\,C_{\mu_{2}\nu^{t}_{1},\alpha^{t}_{1}}(t,q)\,C_{\mu^{t}_{2}\nu_{2}\alpha_{5}}(q,t)\,C_{\mu_{3}\nu_{2}^{t}\alpha^{t}_{2}}(t,q)\,C_{\mu^{t}_{6}\nu_{3}\alpha_{3}}(q,t)\,C_{\mu_{4}\nu^{t}_{3}\alpha^{t}_{3}}(t,q)\,
×Cμ4tν4α1(q,t)Cμ5ν4tα4t(t,q)Cμ5tν5α2(q,t)Cμ6ν5tα5t(t,q)Cμ6tν6α3(q,t)Cμ1ν6tα6t(t,q).absentsubscript𝐶subscriptsuperscript𝜇𝑡4subscript𝜈4subscript𝛼1𝑞𝑡subscript𝐶subscript𝜇5subscriptsuperscript𝜈𝑡4superscriptsubscript𝛼4𝑡𝑡𝑞subscript𝐶superscriptsubscript𝜇5𝑡subscript𝜈5subscript𝛼2𝑞𝑡subscript𝐶subscript𝜇6superscriptsubscript𝜈5𝑡superscriptsubscript𝛼5𝑡𝑡𝑞subscript𝐶subscriptsuperscript𝜇𝑡6subscript𝜈6subscript𝛼3𝑞𝑡subscript𝐶subscript𝜇1superscriptsubscript𝜈6𝑡superscriptsubscript𝛼6𝑡𝑡𝑞\displaystyle\hskip 14.22636pt\times C_{\mu^{t}_{4}\nu_{4}\alpha_{1}}(q,t)\,C_{\mu_{5}\nu^{t}_{4}\alpha_{4}^{t}}(t,q)\,C_{\mu_{5}^{t}\nu_{5}\alpha_{2}}(q,t)\,C_{\mu_{6}\nu_{5}^{t}\alpha_{5}^{t}}(t,q)\,C_{\mu^{t}_{6}\nu_{6}\alpha_{3}}(q,t)\,C_{\mu_{1}\nu_{6}^{t}\alpha_{6}^{t}}(t,q)\,. (III.15)

Information of our conventions regarding integer partitions (and functional identities which shall be important in the following) is compiled in appendix A, and the refined topological vertex Cμνρ(q,t)subscript𝐶𝜇𝜈𝜌𝑞𝑡C_{\mu\nu\rho}(q,t) is defined in (B.1). Using the expression for the latter, we can write (III.15) as

Wα1α2α3α4α5α6α4α5α6α1α2α3(Qhi,Qvi,ϵ1,ϵ2)=(k=16tαk22qαkt22Z~αk(q,t)Z~αkt(t,q))superscriptsubscript𝑊subscript𝛼1subscript𝛼2subscript𝛼3subscript𝛼4subscript𝛼5subscript𝛼6subscript𝛼4subscript𝛼5subscript𝛼6subscript𝛼1subscript𝛼2subscript𝛼3subscript𝑄subscript𝑖subscript𝑄subscript𝑣𝑖subscriptitalic-ϵ1subscriptitalic-ϵ2superscriptsubscriptproduct𝑘16superscript𝑡superscriptnormsubscript𝛼𝑘22superscript𝑞superscriptnormsuperscriptsubscript𝛼𝑘𝑡22subscript~𝑍subscript𝛼𝑘𝑞𝑡subscript~𝑍superscriptsubscript𝛼𝑘𝑡𝑡𝑞\displaystyle W_{\alpha_{1}\,\alpha_{2}\,\alpha_{3}\,\alpha_{4}\,\alpha_{5}\,\alpha_{6}}^{\alpha_{4}\,\alpha_{5}\,\alpha_{6}\,\alpha_{1}\,\alpha_{2}\,\alpha_{3}}(Q_{h_{i}},Q_{v_{i}},\epsilon_{1},\epsilon_{2})=\Big{(}\prod_{k=1}^{6}t^{\frac{||\alpha_{k}||^{2}}{2}}q^{\frac{||\alpha_{k}^{t}||^{2}}{2}}\tilde{Z}_{\alpha_{k}}(q,t)\tilde{Z}_{\alpha_{k}^{t}}(t,q)\Big{)}
×{μ}{ν}{η}{η~}i=16(Qhi)|μi|(Qvi)|νi|sμi/η~i+3(xi+3)sμit/ηi+5(yi+5)sνit/ηi(wi)sνi/η~i+3(zi+3),\displaystyle\hskip 28.45274pt\times\sum_{\begin{subarray}{c}\{\mu\}\{\nu\}\\ \{\eta\}\{\tilde{\eta}\}\end{subarray}}\prod_{i=1}^{6}(-Q_{h_{i}})^{|\mu_{i}|}(-Q_{v_{i}})^{|\nu_{i}|}s_{\mu_{i}/\tilde{\eta}_{i+3}}(x_{i+3})s_{\mu_{i}^{t}/\eta_{i+5}}(y_{i+5})s_{\nu_{i}^{t}/\eta_{i}}(w_{i})s_{\nu_{i}/\tilde{\eta}_{i+3}}(z_{i+3})\,, (III.16)

where sμ/νsubscript𝑠𝜇𝜈s_{\mu/\nu} are (skew) Schur functions (see Macdo for the definition). The partitions in (III.16) are cyclically identified (e.g. ηi+6=ηisubscript𝜂𝑖6subscript𝜂𝑖\eta_{i+6}=\eta_{i}) and variables are defined as

xi=qρ+12tαi12,subscript𝑥𝑖superscript𝑞𝜌12superscript𝑡subscript𝛼𝑖12\displaystyle x_{i}=q^{-\rho+\frac{1}{2}}t^{-\alpha_{i}-\frac{1}{2}}\,, yi=tρ+12qαit12,subscript𝑦𝑖superscript𝑡𝜌12superscript𝑞superscriptsubscript𝛼𝑖𝑡12\displaystyle y_{i}=t^{-\rho+\frac{1}{2}}q^{-\alpha_{i}^{t}-\frac{1}{2}}\,,
wi=qρtαi,subscript𝑤𝑖superscript𝑞𝜌superscript𝑡subscript𝛼𝑖\displaystyle w_{i}=q^{-\rho}t^{-\alpha_{i}}\,, zi=tρqαit.subscript𝑧𝑖superscript𝑡𝜌superscript𝑞superscriptsubscript𝛼𝑖𝑡\displaystyle z_{i}=t^{-\rho}q^{-\alpha_{i}^{t}}\,. (III.17)

Using the identities given in Appendix A, we can write Eq.(III.16) in the form (see Appendix B for details)

Wα1α2α3α4α5α6α4α5α6α1α2α3=W2,3()[(tq)52l=16Qhl]i=16|αi|i,j=16(r,s)αiθ1(ρ,z^r,s(i,j))θ1(ρ,u^r,s(i,j))θ1(ρ,w^r,s(i,j))θ1(ρ,v^r,s(i,j)),superscriptsubscript𝑊subscript𝛼1subscript𝛼2subscript𝛼3subscript𝛼4subscript𝛼5subscript𝛼6subscript𝛼4subscript𝛼5subscript𝛼6subscript𝛼1subscript𝛼2subscript𝛼3subscript𝑊23superscriptdelimited-[]superscript𝑡𝑞52superscriptsubscriptproduct𝑙16subscript𝑄subscript𝑙superscriptsubscript𝑖16subscript𝛼𝑖superscriptsubscriptproduct𝑖𝑗16subscriptproduct𝑟𝑠subscript𝛼𝑖subscript𝜃1𝜌subscriptsuperscript^𝑧𝑖𝑗𝑟𝑠subscript𝜃1𝜌subscriptsuperscript^𝑢𝑖𝑗𝑟𝑠subscript𝜃1𝜌subscriptsuperscript^𝑤𝑖𝑗𝑟𝑠subscript𝜃1𝜌subscriptsuperscript^𝑣𝑖𝑗𝑟𝑠\displaystyle W_{\alpha_{1}\,\alpha_{2}\,\alpha_{3}\,\alpha_{4}\,\alpha_{5}\,\alpha_{6}}^{\alpha_{4}\,\alpha_{5}\,\alpha_{6}\,\alpha_{1}\,\alpha_{2}\,\alpha_{3}}=W_{2,3}(\emptyset)\,\left[\left(\frac{t}{q}\right)^{\frac{5}{2}}\prod_{l=1}^{6}Q_{h_{l}}\right]^{\sum_{i=1}^{6}|\alpha_{i}|}\prod_{i,j=1}^{6}\prod_{(r,s)\in\alpha_{i}}\frac{\theta_{1}(\rho,\hat{z}^{(i,j)}_{r,s})\theta_{1}(\rho,\hat{u}^{(i,j)}_{r,s})}{\theta_{1}(\rho,\hat{w}^{(i,j)}_{r,s})\theta_{1}(\rho,\hat{v}^{(i,j)}_{r,s})}\,, (III.18)

where θ1subscript𝜃1\theta_{1} is the Jacobi-theta function

θ1subscript𝜃1\displaystyle\theta_{1} (ρ;z)=ieiπτ4eiπzk=1(1e2πikρ)(1e2πikρe2πiz)(1e2πi(k1)ρe2πiz),𝜌𝑧𝑖superscript𝑒𝑖𝜋𝜏4superscript𝑒𝑖𝜋𝑧superscriptsubscriptproduct𝑘11superscript𝑒2𝜋𝑖𝑘𝜌1superscript𝑒2𝜋𝑖𝑘𝜌superscript𝑒2𝜋𝑖𝑧1superscript𝑒2𝜋𝑖𝑘1𝜌superscript𝑒2𝜋𝑖𝑧\displaystyle(\rho;z)=-ie^{\frac{i\pi\tau}{4}}e^{i\pi z}\prod_{k=1}^{\infty}(1-e^{2\pi ik\rho})\,(1-e^{2\pi ik\rho}e^{2\pi iz})(1-e^{2\pi i(k-1)\rho}e^{-2\pi iz})\,,

and W2,3()=W6()=Wsubscript𝑊23subscript𝑊6subscriptsuperscript𝑊W_{2,3}(\emptyset)=W_{6}(\emptyset)=W^{\emptyset\emptyset\emptyset\emptyset\emptyset\emptyset}_{\emptyset\emptyset\emptyset\emptyset\emptyset\emptyset} is a normalisation factor (see Haghighat:2013gba ; Hohenegger:2013ala and Eq.(IV.11) for L=6𝐿6L=6 below). Furthermore, the arguments of theta functions are given by

z^r,s(i,j)=βi+3,j2πi+ϵ1(αi+3j,str+12)ϵ2(αi,rs+12),subscriptsuperscript^𝑧𝑖𝑗𝑟𝑠subscript𝛽𝑖3𝑗2𝜋𝑖subscriptitalic-ϵ1superscriptsubscript𝛼𝑖3𝑗𝑠𝑡𝑟12subscriptitalic-ϵ2subscript𝛼𝑖𝑟𝑠12\displaystyle\hat{z}^{(i,j)}_{r,s}=\frac{\beta_{i+3,j}}{2\pi i}+\epsilon_{1}(\alpha_{i+3-j,s}^{t}-r+\tfrac{1}{2})-\epsilon_{2}(\alpha_{i,r}-s+\tfrac{1}{2})\,,
u^r,s(i,j)=βi+j,j2πiϵ1(αi+j3,str+12)+ϵ2(αi,rs+12),subscriptsuperscript^𝑢𝑖𝑗𝑟𝑠subscript𝛽𝑖𝑗𝑗2𝜋𝑖subscriptitalic-ϵ1superscriptsubscript𝛼𝑖𝑗3𝑠𝑡𝑟12subscriptitalic-ϵ2subscript𝛼𝑖𝑟𝑠12\displaystyle\hat{u}^{(i,j)}_{r,s}=\frac{\beta_{i+j,j}}{2\pi i}-\epsilon_{1}(\alpha_{i+j-3,s}^{t}-r+\tfrac{1}{2})+\epsilon_{2}(\alpha_{i,r}-s+\tfrac{1}{2})\,,
w^r,s(i,j)=λi+3,j2πi+ϵ1(αij,str)ϵ2(αi,rs+1),subscriptsuperscript^𝑤𝑖𝑗𝑟𝑠subscript𝜆𝑖3𝑗2𝜋𝑖subscriptitalic-ϵ1superscriptsubscript𝛼𝑖𝑗𝑠𝑡𝑟subscriptitalic-ϵ2subscript𝛼𝑖𝑟𝑠1\displaystyle\hat{w}^{(i,j)}_{r,s}=\frac{\lambda_{i+3,j}}{2\pi i}+\epsilon_{1}(\alpha_{i-j,s}^{t}-r)-\epsilon_{2}(\alpha_{i,r}-s+1)\,,
v^r,s(i,j)=λi+j3,j2πiϵ1(αi+j,str+1)+ϵ2(αi,rs)subscriptsuperscript^𝑣𝑖𝑗𝑟𝑠subscript𝜆𝑖𝑗3𝑗2𝜋𝑖subscriptitalic-ϵ1superscriptsubscript𝛼𝑖𝑗𝑠𝑡𝑟1subscriptitalic-ϵ2subscript𝛼𝑖𝑟𝑠\displaystyle\hat{v}^{(i,j)}_{r,s}=\frac{\lambda_{i+j-3,j}}{2\pi i}-\epsilon_{1}(\alpha_{i+j,s}^{t}-r+1)+\epsilon_{2}(\alpha_{i,r}-s)\,

with the shorthand notation

βi,j=hi+k=1j1(hik+vik),subscript𝛽𝑖𝑗subscript𝑖superscriptsubscript𝑘1𝑗1subscript𝑖𝑘subscript𝑣𝑖𝑘\displaystyle\beta_{i,j}=h_{i}+\sum_{k=1}^{j-1}(h_{i-k}+v_{i-k})\,,
λi,j={0if j=6hi+k=1j1(hik+vik)+vijelsesubscript𝜆𝑖𝑗cases0if 𝑗6subscript𝑖superscriptsubscript𝑘1𝑗1subscript𝑖𝑘subscript𝑣𝑖𝑘subscript𝑣𝑖𝑗else\displaystyle\lambda_{i,j}=\begin{cases}0&\quad\text{if }j=6\\ h_{i}+\sum_{k=1}^{j-1}(h_{i-k}+v_{i-k})+v_{i-j}&\quad\text{else}\\ \end{cases}

III.5 Comparison of 𝒵6,1subscript𝒵61\mathcal{Z}_{6,1} with 𝒵3,2subscript𝒵32\mathcal{Z}_{3,2}

In order to verify the relation (III.1), we have to compare (III.13) with (III.10). To this end, we first express the arguments of θ1subscript𝜃1\theta_{1}-functions in (III.18) in terms of the parameters (V,M,Hi)𝑉𝑀subscript𝐻𝑖(V,M,H_{i}) and using Eq. (III.9):

βi,1=MHi1Hi,subscript𝛽𝑖1𝑀subscript𝐻𝑖1subscript𝐻𝑖\displaystyle\beta_{i,1}=-M-H_{i-1}-H_{i}\,, βi,4=2M+Hi2,subscript𝛽𝑖42𝑀subscript𝐻𝑖2\displaystyle\beta_{i,4}=2M+H_{i-2}\,,
βi,2=Hi1,subscript𝛽𝑖2subscript𝐻𝑖1\displaystyle\beta_{i,2}=-H_{i-1}\,, βi,5=3M+Hi3+Hi2,subscript𝛽𝑖53𝑀subscript𝐻𝑖3subscript𝐻𝑖2\displaystyle\beta_{i,5}=3M+H_{i-3}+H_{i-2}\,,
βi,3=M,subscript𝛽𝑖3𝑀\displaystyle\beta_{i,3}=M\,, βi,6=4M+r=13Hi1r,subscript𝛽𝑖64𝑀superscriptsubscript𝑟13subscript𝐻𝑖1𝑟\displaystyle\beta_{i,6}=4M+\sum_{r=1}^{3}H_{i-1-r}\,,

as well as

λi,j={0ifj=6jM+Hi2++Hi1jelse.subscript𝜆𝑖𝑗cases0if𝑗6𝑗𝑀subscript𝐻𝑖2subscript𝐻𝑖1𝑗else.\lambda_{i,j}=\begin{cases}0&\quad\text{if}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ j=6\\ jM+H_{i-2}+\dots+H_{i-1-j}&\quad\text{else.}\end{cases}

The prefactor in Eq. (III.18) then becomes

[(tq)52l=16Qhl]i=16|αi|=[(tq)52Qρ1i=16QHi1]i=16|αi|.superscriptdelimited-[]superscript𝑡𝑞52superscriptsubscriptproduct𝑙16subscript𝑄subscript𝑙superscriptsubscript𝑖16subscript𝛼𝑖superscriptdelimited-[]superscript𝑡𝑞52superscriptsubscript𝑄𝜌1superscriptsubscriptproduct𝑖16superscriptsubscript𝑄subscript𝐻𝑖1superscriptsubscript𝑖16subscript𝛼𝑖\displaystyle\Big{[}\Big{(}\frac{t}{q}\Big{)}^{\frac{5}{2}}\prod_{l=1}^{6}Q_{h_{l}}\Big{]}^{\sum_{i=1}^{6}|\alpha_{i}|}=\Big{[}\Big{(}\frac{t}{q}\Big{)}^{\frac{5}{2}}Q_{\rho}^{-1}\prod_{i=1}^{6}Q_{H_{i}}^{-1}\Big{]}^{\sum_{i=1}^{6}|\alpha_{i}|}\,.

Furthermore, we can change the arguments of the θ1subscript𝜃1\theta_{1}-functions in Eq. (III.15) using the following identity of Jacobi theta functions under the shift by an integer n𝑛n:

θ1(ρ;z+nρ)=Qρn22(e2πiz)nθ1(ρ;z).subscript𝜃1𝜌𝑧𝑛𝜌superscriptsubscript𝑄𝜌superscript𝑛22superscriptsuperscript𝑒2𝜋𝑖𝑧𝑛subscript𝜃1𝜌𝑧\displaystyle\theta_{1}\left(\rho;z+n\,\rho\right)=Q_{\rho}^{-\frac{n^{2}}{2}}\,(-e^{-2\pi iz})^{n}\,\theta_{1}(\rho;z)\,. (III.19)

In this manner, Eq. (III.15) can be written as

Wα1α2α3α4α5α6α4α5α6α1α2α3(QHi,QM,ϵ1,ϵ2)=[(tq)5/2Qρ1i=16QHi1]i=16|αi|[(qt)5/2]i=16|αi|superscriptsubscript𝑊subscript𝛼1subscript𝛼2subscript𝛼3subscript𝛼4subscript𝛼5subscript𝛼6subscript𝛼4subscript𝛼5subscript𝛼6subscript𝛼1subscript𝛼2subscript𝛼3subscript𝑄subscript𝐻𝑖subscript𝑄𝑀subscriptitalic-ϵ1subscriptitalic-ϵ2superscriptdelimited-[]superscript𝑡𝑞52superscriptsubscript𝑄𝜌1superscriptsubscriptproduct𝑖16superscriptsubscript𝑄subscript𝐻𝑖1superscriptsubscript𝑖16subscript𝛼𝑖superscriptdelimited-[]superscript𝑞𝑡52superscriptsubscript𝑖16subscript𝛼𝑖\displaystyle W_{\alpha_{1}\,\alpha_{2}\,\alpha_{3}\,\alpha_{4}\,\alpha_{5}\,\alpha_{6}}^{\alpha_{4}\,\alpha_{5}\,\alpha_{6}\,\alpha_{1}\,\alpha_{2}\,\alpha_{3}}(Q_{H_{i}},Q_{M},\epsilon_{1},\epsilon_{2})=\left[\left(\frac{t}{q}\right)^{5/2}Q_{\rho}^{-1}\prod_{i=1}^{6}Q_{H_{i}}^{-1}\right]^{\sum_{i=1}^{6}|\alpha_{i}|}\,\Big{[}-\Big{(}\frac{q}{t}\Big{)}^{5/2}\Big{]}^{\sum_{i=1}^{6}|\alpha_{i}|}
×(Qρ2QM7QH11QH22QH32QH41)|α1|(Qρ2QM7QH21QH32QH42QH51)|α2|(QρQM1QH1QH2QH41QH51)|α3|absentsuperscriptsubscriptsuperscript𝑄2𝜌superscriptsubscript𝑄𝑀7superscriptsubscript𝑄subscript𝐻11superscriptsubscript𝑄subscript𝐻22superscriptsubscript𝑄subscript𝐻32superscriptsubscript𝑄subscript𝐻41subscript𝛼1superscriptsubscriptsuperscript𝑄2𝜌superscriptsubscript𝑄𝑀7superscriptsubscript𝑄subscript𝐻21superscriptsubscript𝑄subscript𝐻32superscriptsubscript𝑄subscript𝐻42superscriptsubscript𝑄subscript𝐻51subscript𝛼2superscriptsubscript𝑄𝜌superscriptsubscript𝑄𝑀1subscript𝑄subscript𝐻1subscript𝑄subscript𝐻2superscriptsubscript𝑄subscript𝐻41superscriptsubscript𝑄subscript𝐻51subscript𝛼3\displaystyle\hskip 42.67912pt\times(Q^{2}_{\rho}Q_{M}^{-7}Q_{H_{1}}^{-1}Q_{H_{2}}^{-2}Q_{H_{3}}^{-2}Q_{H_{4}}^{-1})^{|\alpha_{1}|}\,(Q^{2}_{\rho}Q_{M}^{-7}Q_{H_{2}}^{-1}Q_{H_{3}}^{-2}Q_{H_{4}}^{-2}Q_{H_{5}}^{-1})^{|\alpha_{2}|}(Q_{\rho}Q_{M}^{-1}Q_{H_{1}}Q_{H_{2}}Q_{H_{4}}^{-1}Q_{H_{5}}^{-1})^{|\alpha_{3}|}
×(QM5QH1QH22QH32QH4)|α4|(QM5QH2QH32QH42QH5)|α5|(QρQM1QH11QH21QH4QH5)|α6|absentsuperscriptsuperscriptsubscript𝑄𝑀5subscript𝑄subscript𝐻1superscriptsubscript𝑄subscript𝐻22superscriptsubscript𝑄subscript𝐻32subscript𝑄subscript𝐻4subscript𝛼4superscriptsuperscriptsubscript𝑄𝑀5subscript𝑄subscript𝐻2superscriptsubscript𝑄subscript𝐻32superscriptsubscript𝑄subscript𝐻42subscript𝑄subscript𝐻5subscript𝛼5superscriptsubscript𝑄𝜌superscriptsubscript𝑄𝑀1superscriptsubscript𝑄subscript𝐻11superscriptsubscript𝑄subscript𝐻21subscript𝑄subscript𝐻4subscript𝑄subscript𝐻5subscript𝛼6\displaystyle\hskip 42.67912pt\times(Q_{M}^{5}Q_{H_{1}}Q_{H_{2}}^{2}Q_{H_{3}}^{2}Q_{H_{4}})^{|\alpha_{4}|}(Q_{M}^{5}Q_{H_{2}}Q_{H_{3}}^{2}Q_{H_{4}}^{2}Q_{H_{5}})^{|\alpha_{5}|}(Q_{\rho}Q_{M}^{-1}Q_{H_{1}}^{-1}Q_{H_{2}}^{-1}Q_{H_{4}}Q_{H_{5}})^{|\alpha_{6}|}
×(a=16ϑαaαa(QM;ρ)ϑαaαa(q/t;ρ))(1a<b6ϑαaαb(QabQM;ρ)ϑαaαb(QabQM1;ρ)ϑαaαb(Qabt/q;ρ)ϑαaαb(Qabq/t;ρ)),absentsuperscriptsubscriptproduct𝑎16subscriptitalic-ϑsubscript𝛼𝑎subscript𝛼𝑎subscript𝑄𝑀𝜌subscriptitalic-ϑsubscript𝛼𝑎subscript𝛼𝑎𝑞𝑡𝜌subscriptproduct1𝑎𝑏6subscriptitalic-ϑsubscript𝛼𝑎subscript𝛼𝑏subscript𝑄𝑎𝑏subscript𝑄𝑀𝜌subscriptitalic-ϑsubscript𝛼𝑎subscript𝛼𝑏subscript𝑄𝑎𝑏superscriptsubscript𝑄𝑀1𝜌subscriptitalic-ϑsubscript𝛼𝑎subscript𝛼𝑏subscript𝑄𝑎𝑏𝑡𝑞𝜌subscriptitalic-ϑsubscript𝛼𝑎subscript𝛼𝑏subscript𝑄𝑎𝑏𝑞𝑡𝜌\displaystyle\hskip 42.67912pt\times\left(\prod_{a=1}^{6}\frac{\vartheta_{\alpha_{a}\alpha_{a}}(Q_{M};\rho)}{\vartheta_{\alpha_{a}\alpha_{a}}(\sqrt{q/t};\rho)}\right)\,\left(\prod_{1\leq a<b\leq 6}\frac{\vartheta_{\alpha_{a}\alpha_{b}}(Q_{ab}Q_{M};\rho)\vartheta_{\alpha_{a}\alpha_{b}}(Q_{ab}Q_{M}^{-1};\rho)}{\vartheta_{\alpha_{a}\alpha_{b}}(Q_{ab}\sqrt{t/q};\rho)\vartheta_{\alpha_{a}\alpha_{b}}(Q_{ab}\sqrt{q/t};\rho)}\right)\,, (III.20)

where we combined the θ1subscript𝜃1\theta_{1}-functions into ϑitalic-ϑ\vartheta-functions which are defined in Appendix A (using also Eq. (A.9)). Thus, the whole partition function 𝒵3,2(V,M,H1,,6,ϵ1,2)subscript𝒵32𝑉𝑀subscript𝐻16subscriptitalic-ϵ12\mathcal{Z}_{3,2}(V,M,H_{1,\ldots,6},\epsilon_{1,2}) in Eq. (III.13) becomes

𝒵X3,2(V,M,H1,,6,ϵ1,ϵ2)subscript𝒵subscript𝑋32𝑉𝑀subscript𝐻16subscriptitalic-ϵ1subscriptitalic-ϵ2\displaystyle{\cal Z}_{X_{3,2}}(V,M,H_{1,\ldots,6},\epsilon_{1},\epsilon_{2}) =W2,3()×α1,,α6(QVQM)|α1|++|α6|(a=16ϑαaαa(QM;ρ)ϑαaαa(q/t;ρ))absentsubscript𝑊23subscriptsubscript𝛼1subscript𝛼6superscriptsubscript𝑄𝑉subscript𝑄𝑀subscript𝛼1subscript𝛼6superscriptsubscriptproduct𝑎16subscriptitalic-ϑsubscript𝛼𝑎subscript𝛼𝑎subscript𝑄𝑀𝜌subscriptitalic-ϑsubscript𝛼𝑎subscript𝛼𝑎𝑞𝑡𝜌\displaystyle=W_{2,3}(\emptyset)\times\sum_{\alpha_{1},\ldots,\alpha_{6}}(-Q_{V}Q_{M})^{|\alpha_{1}|+\dots+|\alpha_{6}|}\left(\prod_{a=1}^{6}\frac{\vartheta_{\alpha_{a}\alpha_{a}}(Q_{M};\rho)}{\vartheta_{\alpha_{a}\alpha_{a}}(\sqrt{q/t};\rho)}\right)
×(1a<b6ϑαaαb(QabQM;ρ)ϑαaαb(QabQM1;ρ)ϑαaαb(Qabt/q;ρ)ϑαaαb(Qabq/t;ρ)).absentsubscriptproduct1𝑎𝑏6subscriptitalic-ϑsubscript𝛼𝑎subscript𝛼𝑏subscript𝑄𝑎𝑏subscript𝑄𝑀𝜌subscriptitalic-ϑsubscript𝛼𝑎subscript𝛼𝑏subscript𝑄𝑎𝑏superscriptsubscript𝑄𝑀1𝜌subscriptitalic-ϑsubscript𝛼𝑎subscript𝛼𝑏subscript𝑄𝑎𝑏𝑡𝑞𝜌subscriptitalic-ϑsubscript𝛼𝑎subscript𝛼𝑏subscript𝑄𝑎𝑏𝑞𝑡𝜌\displaystyle\times\left(\prod_{1\leq a<b\leq 6}\frac{\vartheta_{\alpha_{a}\alpha_{b}}(Q_{ab}Q_{M};\rho)\vartheta_{\alpha_{a}\alpha_{b}}(Q_{ab}Q_{M}^{-1};\rho)}{\vartheta_{\alpha_{a}\alpha_{b}}(Q_{ab}\sqrt{t/q};\rho)\vartheta_{\alpha_{a}\alpha_{b}}(Q_{ab}\sqrt{q/t};\rho)}\right)\,. (III.21)

We see that this agrees with the partition function 𝒵6,1(V,M,Hi,ϵ1,2)subscript𝒵61𝑉𝑀subscript𝐻𝑖subscriptitalic-ϵ12\mathcal{Z}_{6,1}(V,M,H_{i},\epsilon_{1,2}) in Eq. (III.10). This proves (III.1) at a generic point in the Kähler moduli space and for generic values of ϵ1,2subscriptitalic-ϵ12\epsilon_{1,2}. This result provides a very strong support to the duality proposed in Hohenegger:2016yuv .

IV Building Block for General (N,M)𝑁𝑀(N,M) Web

In the last section, we computed the partition function 𝒵3,2subscript𝒵32\mathcal{Z}_{3,2} at a generic point in the moduli space by choosing the preferred direction of the (refined) topological vertex along the diagonal. As we have seen, this leads to a series representation of 𝒵3,2subscript𝒵32\mathcal{Z}_{3,2} that was instrumental in proving the duality (III.1). In this section, we generalise this result to generic configurations of the type (N,M)𝑁𝑀(N,M). To this end, we first discuss the ’diagonal expansion’ for generic (N,M)𝑁𝑀(N,M) and, in a second step, derive a building block (the generalisation of W𝑊W in (III.13)) which allows the computation of 𝒵N,Msubscript𝒵𝑁𝑀\mathcal{Z}_{N,M} in full generality.

IV.1 Diagonal Expansion of a Generic (N,M)𝑁𝑀(N,M) Web

The most direct way to understand generalisation of the diagonal expansion for a generic (N,M)𝑁𝑀(N,M) web is from the perspective of dual Newton polygon: if the external legs of the web in Fig. 1 were not glued together (i.e. mutually identified), the corresponding Newton polygon would be an N×M𝑁𝑀N\times M rectangle in which each 1×1111\times 1 square is triangulated in the same way (as all the diagonal intervals are parallel). The gluing of external lines creates a periodic web, whose Newton polygon can be obtained by drawing the N×M𝑁𝑀N\times M rectangle on the torus (i.e. gluing the parallel edges of N×M𝑁𝑀N\times M polygon). We can, however, also consider the periodic Newton polygon as a tiling of the plane in terms of basic N×M𝑁𝑀N\times M rectangle. This is shown for the case (3,2)32(3,2) in Fig. 2(b) and more generically in Fig. 6(a). In both cases, the grey region is the N×M𝑁𝑀N\times M rectangle that is dual to the web diagram of XN,Msubscript𝑋𝑁𝑀X_{N,M}.

As in the (3,2)32(3,2) case, however, the fundamental domain of a given tiling is not unique. Indeed, the green region in Fig. 6(a) also contains all distinct faces of the Newton polygon, and is therefore a possible choice of the fundamental domain. However, the web diagram dual to this green region (drawn in Fig. 6(b)) shows that XN,Msubscript𝑋𝑁𝑀X_{N,M} can also be drawn as a (NMk,k)𝑁𝑀𝑘𝑘(\frac{NM}{k},k) web (with k=gcd(N,M)𝑘gcd𝑁𝑀k=\text{gcd}(N,M)) but not with ‘adjacent’ external legs on opposite sides glued together. Indeed, the ‘off-set’ Δ=δ/NΔ𝛿𝑁\Delta=\delta/N can be read off by comparing the positions of the two red intervals in Fig. 6(a) and is given by the minimal (integer) solution to the diophantine equation

δ=ΔN=nMk,forn.formulae-sequence𝛿Δ𝑁𝑛𝑀𝑘for𝑛\displaystyle\displaystyle\delta=\Delta N=nM-k\,,\leavevmode\nobreak\ \leavevmode\nobreak\ \text{for}\leavevmode\nobreak\ \leavevmode\nobreak\ n\in\mathbb{Z}\,. (IV.1)

We will call generic web diagrams with δ0𝛿0\delta\neq 0 twisted, the twisted (6,1)61(6,1) diagram being an example already encountered in the last section. Note, however, that twisted webs of the type Fig. 6(b) can also be brought to the form of an untwisted (NMk,k)𝑁𝑀𝑘𝑘(\frac{NM}{k},k) web by a series of flop transitions Hohenegger:2016yuv . It turns out, however, that the presentation of the (N,M)𝑁𝑀(N,M) web diagram as a twisted (NMk,k)𝑁𝑀𝑘𝑘(\frac{NM}{k},k) web is more suited for the computation of the partition function.

k𝑘kNMk𝑁𝑀𝑘\frac{NM}{k}δ𝛿\deltaM𝑀MN𝑁N(a)(a)
\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdotsa1subscript𝑎1a_{1}a2subscript𝑎2a_{2}ak1subscript𝑎𝑘1a_{k-1}aksubscript𝑎𝑘a_{k}a1subscript𝑎1a_{1}a2subscript𝑎2a_{2}ak1subscript𝑎𝑘1a_{k-1}aksubscript𝑎𝑘a_{k}111222ΔΔ\DeltaΔ+1Δ1\Delta+1NMk1𝑁𝑀𝑘1\tfrac{NM}{k}-1NMk𝑁𝑀𝑘\tfrac{NM}{k}NMkΔ𝑁𝑀𝑘Δ\tfrac{NM}{k}-\DeltaNMk+1Δ𝑁𝑀𝑘1Δ\tfrac{NM}{k}+1-\DeltaNMk𝑁𝑀𝑘\tfrac{NM}{k}111Δ1Δ1\Delta-1ΔΔ\Delta(b)
Figure 6: (a) Newton polygon of XN,Msubscript𝑋𝑁𝑀X_{N,M} (for better readability we have chosen (N,M)=(6,4)𝑁𝑀64(N,M)=(6,4)): The grey and green region highlight two different (but equivalent) fundamental domains that are used to tile the plane. Indeed, the grey region is the dual polygon of the web shown in Fig. 1, while the dual of the green region is the twisted web diagram. The distance δ𝛿\delta between the two equivalent intervals colored in red determines the shift ΔΔ\Delta (for the particular case (6,4)64(6,4) we have δ=6=N𝛿6𝑁\delta=6=N and Δ=1Δ1\Delta=1). (b) Twisted web diagram for generic (N,M)𝑁𝑀(N,M). The orange region highlights one of the building blocks which can be used to compute the partition function 𝒵N,Msubscript𝒵𝑁𝑀\mathcal{Z}_{N,M} and which is shown in more detail in Fig. 7 along with a labelling of the relevant parameters and integer partitions in preparation of the topological vertex computation.

IV.2 Generic Building Block

The twisted web diagram is decomposable into k=gcd(N,M)𝑘gcd𝑁𝑀k=\text{gcd}(N,M) basic strips (one of them is highlighted in orange color in Fig. 6) which are glued together along the diagonal intervals. We refer to these strips sometimes as ‘staircase’ diagrams and a generic such strip of length L𝐿L (along with a suitable labelling of Kähler parameters associated with the various intervals, as well as integer partitions associated with the external diagonal legs) is shown in Fig. 7. The open string amplitudes for such strips can be glued together to form the topological string partition function of XN,Msubscript𝑋𝑁𝑀X_{N,M} (see Haghighat:2013gba for the case XN,1subscript𝑋𝑁1X_{N,1}, Hohenegger:2013ala for XN,Msubscript𝑋𝑁𝑀X_{N,M} with a non-maximal set of Kähler parameters and Haghighat:2013tka for the limit in which one of the elliptic fibrations of XN,Msubscript𝑋𝑁𝑀X_{N,M} degenerates). Below, we compute the generic building block associated with the strip in Fig. 7 at a generic point in moduli space, which allows to evaluate 𝒵N,Msubscript𝒵𝑁𝑀\mathcal{Z}_{N,M} at an arbitrary point in the Kähler moduli space.

\cdotsα1subscript𝛼1\alpha_{1}α2subscript𝛼2\alpha_{2}α3subscript𝛼3\alpha_{3}β1tsubscriptsuperscript𝛽𝑡1\beta^{t}_{1}β2tsubscriptsuperscript𝛽𝑡2\beta^{t}_{2}a𝑎ah1,μ1subscript1subscript𝜇1h_{1},\mu_{1}h2,μ2subscript2subscript𝜇2h_{2},\mu_{2}h3,μ3subscript3subscript𝜇3h_{3},\mu_{3}v1,ν1subscript𝑣1subscript𝜈1v_{1},\nu_{1}v2,ν2subscript𝑣2subscript𝜈2v_{2},\nu_{2}vL,νLsubscript𝑣𝐿subscript𝜈𝐿v_{L},\nu_{L}hL,μLsubscript𝐿subscript𝜇𝐿h_{L},\mu_{L}h1,μ1subscript1subscript𝜇1h_{1},\mu_{1}βL1tsubscriptsuperscript𝛽𝑡𝐿1\beta^{t}_{L-1}βLtsubscriptsuperscript𝛽𝑡𝐿\beta^{t}_{L}αLsubscript𝛼𝐿\alpha_{L}a𝑎aa^1subscript^𝑎1\widehat{a}_{1}a^2subscript^𝑎2\widehat{a}_{2}\cdotsa^Lsubscript^𝑎𝐿\widehat{a}_{L}b^1subscript^𝑏1\widehat{b}_{1}b^2subscript^𝑏2\widehat{b}_{2}\cdotsb^Lsubscript^𝑏𝐿\widehat{b}_{L}
Figure 7: ‘Staircase’ strip of length L𝐿L with a labelling of the Kähler parameters and integer partitions.

Using the refined topological vertex, we can express the generic building block for the strip in Fig. 7 as

Wβ1βLα1αL(Qhi,Qvi,ϵ1,ϵ2)=Z^×{μ}{ν}{η}{η~}i=1LQhi|μi|Qvi|νi|sμi/ηi(xi)sμit/η~i1(yi1)sνit/η~i(wi)sνi/ηi(zi)subscriptsuperscript𝑊subscript𝛼1subscript𝛼𝐿subscript𝛽1subscript𝛽𝐿subscript𝑄subscript𝑖subscript𝑄subscript𝑣𝑖subscriptitalic-ϵ1subscriptitalic-ϵ2^𝑍subscript𝜇𝜈𝜂~𝜂superscriptsubscriptproduct𝑖1𝐿superscriptsubscript𝑄subscript𝑖subscript𝜇𝑖superscriptsubscript𝑄subscript𝑣𝑖subscript𝜈𝑖subscript𝑠subscript𝜇𝑖subscript𝜂𝑖subscript𝑥𝑖subscript𝑠superscriptsubscript𝜇𝑖𝑡subscript~𝜂𝑖1subscript𝑦𝑖1subscript𝑠superscriptsubscript𝜈𝑖𝑡subscript~𝜂𝑖subscript𝑤𝑖subscript𝑠subscript𝜈𝑖subscript𝜂𝑖subscript𝑧𝑖\displaystyle W^{\alpha_{1}\dots\alpha_{L}}_{\beta_{1}\dots\beta_{L}}(Q_{h_{i}},Q_{v_{i}},\epsilon_{1},\epsilon_{2})=\hat{Z}\times\sum_{\begin{subarray}{c}\{\mu\}\{\nu\}\\ \{\eta\}\{\tilde{\eta}\}\end{subarray}}\prod_{i=1}^{L}Q_{h_{i}}^{|\mu_{i}|}Q_{v_{i}}^{|\nu_{i}|}s_{\mu_{i}/\eta_{i}}(x_{i})s_{\mu_{i}^{t}/\tilde{\eta}_{i-1}}(y_{i-1})s_{\nu_{i}^{t}/\tilde{\eta}_{i}}(w_{i})s_{\nu_{i}/\eta_{i}}(z_{i}) (IV.2)

where the prefactor is given by

Z^=i=1Ltαk22qαkt22Z~αk(q,t)Z~αkt(t,q),^𝑍superscriptsubscriptproduct𝑖1𝐿superscript𝑡superscriptnormsubscript𝛼𝑘22superscript𝑞superscriptnormsuperscriptsubscript𝛼𝑘𝑡22subscript~𝑍subscript𝛼𝑘𝑞𝑡subscript~𝑍superscriptsubscript𝛼𝑘𝑡𝑡𝑞\hat{Z}=\prod_{i=1}^{L}t^{\frac{||\alpha_{k}||^{2}}{2}}q^{\frac{||\alpha_{k}^{t}||^{2}}{2}}\tilde{Z}_{\alpha_{k}}(q,t)\tilde{Z}_{\alpha_{k}^{t}}(t,q)\,, (IV.3)

and the arguments of skew Schur functions are defined as

xi=qξ+12tαi12,subscript𝑥𝑖superscript𝑞𝜉12superscript𝑡subscript𝛼𝑖12\displaystyle x_{i}=q^{-\xi+\frac{1}{2}}t^{-\alpha_{i}-\frac{1}{2}}\,, yi=tξ+12qβit12,subscript𝑦𝑖superscript𝑡𝜉12superscript𝑞superscriptsubscript𝛽𝑖𝑡12\displaystyle y_{i}=t^{-\xi+\frac{1}{2}}q^{-\beta_{i}^{t}-\frac{1}{2}}\,,
wi=qξtβi,subscript𝑤𝑖superscript𝑞𝜉superscript𝑡subscript𝛽𝑖\displaystyle w_{i}=q^{-\xi}t^{-\beta_{i}}\,, zi=tξqαit,subscript𝑧𝑖superscript𝑡𝜉superscript𝑞superscriptsubscript𝛼𝑖𝑡\displaystyle z_{i}=t^{-\xi}q^{-\alpha_{i}^{t}}\,, (IV.4)

where ξ={12,32,52,}𝜉123252\xi=\left\{-\tfrac{1}{2}\,,-\tfrac{3}{2}\,,-\tfrac{5}{2}\,,\ldots\right\}. It is important to recall that we associate arbitrary (and independent) partitions α1,,Lsubscript𝛼1𝐿\alpha_{1,\ldots,L} and β1,,Lsubscript𝛽1𝐿\beta_{1,\ldots,L} with the upper and lower diagonal legs of the staircase strip (as shown in Fig. 7). Put differently, the diagonal line segments of this staircase are not glued together and therefore no consistency conditions are imposed on the Kähler parameters. Thus, the 2L2𝐿2L variables

h1,,L,subscript1𝐿\displaystyle h_{1,\ldots,L}\,, and v1,,L,subscript𝑣1𝐿\displaystyle v_{1,\ldots,L}\,, (IV.5)

are independent of each other (in the following, we will use the same notation as in (III.14)). Note, however, that this will be changed in section IV.3, when we consider a specific gluing of the external lines of the staircase, as shown in Fig. 9. For later convenience, we also introduce the parameters

a^i=hi+1+visubscript^𝑎𝑖subscript𝑖1subscript𝑣𝑖\displaystyle\widehat{a}_{i}=h_{i+1}+v_{i} and b^i=hi+vi,subscript^𝑏𝑖subscript𝑖subscript𝑣𝑖\displaystyle\widehat{b}_{i}=h_{i}+v_{i}\,, (IV.6)

for all i=1,,L𝑖1𝐿i=1,\ldots,L. These are not all independent of each other, but satisfy the consistency condition i=1La^i=i=1Lb^isuperscriptsubscript𝑖1𝐿subscript^𝑎𝑖superscriptsubscript𝑖1𝐿subscript^𝑏𝑖\sum_{i=1}^{L}\widehat{a}_{i}=\sum_{i=1}^{L}\widehat{b}_{i}.

Comparing with the case (N,M)=(3,2)𝑁𝑀32(N,M)=(3,2) as discussed in the previous section, we note that the structure of (IV.2) is very similar to (III.16), except that the summation involves a larger number of integer partitions. We remark that physically, the sizes of the integer partitions labelling the fundamental building block Wβ1βLα1αLsubscriptsuperscript𝑊subscript𝛼1subscript𝛼𝐿subscript𝛽1subscript𝛽𝐿W^{\alpha_{1}\dots\alpha_{L}}_{\beta_{1}\dots\beta_{L}} give the instanton charges of individual U(1)𝑈1U(1)’s in the dual gauge theory. The dual gauge theory and the role of these partitions labelling the building block will be discussed in detail in paper2 . However, as any given summand in Eq.(IV.2) still contains products of skew Schur functions which can be manipulated using the relations (A.7), we can follow the same steps as in appendix B to work out (IV.2). Specifically, due to the identification of the horizontal ends of the strip (denoted by a𝑎a in Fig. 7), we can generalise the recursion relation (B.6) for generic L𝐿L, which allows us to write a product representation of (IV.2). As the computation is lengthy and very tedious, we refrain from giving the details of various steps and simply state the result in the following form

Wβ1βLα1αLZ^i,j=1Lk,r,s=1(1Q^i,jQρk1xi,ryij,s)(1Q^i,jvQρk1zi,swi+j1,r)(1Q¯i,jQρk1xi,rzij,s)(1Q˙i,jQρk1yi,swi+j,r),similar-tosubscriptsuperscript𝑊subscript𝛼1subscript𝛼𝐿subscript𝛽1subscript𝛽𝐿^𝑍superscriptsubscriptproduct𝑖𝑗1𝐿superscriptsubscriptproduct𝑘𝑟𝑠11subscript^𝑄𝑖𝑗superscriptsubscript𝑄𝜌𝑘1subscript𝑥𝑖𝑟subscript𝑦𝑖𝑗𝑠1subscriptsuperscript^𝑄𝑣𝑖𝑗superscriptsubscript𝑄𝜌𝑘1subscript𝑧𝑖𝑠subscript𝑤𝑖𝑗1𝑟1subscript¯𝑄𝑖𝑗superscriptsubscript𝑄𝜌𝑘1subscript𝑥𝑖𝑟subscript𝑧𝑖𝑗𝑠1subscript˙𝑄𝑖𝑗superscriptsubscript𝑄𝜌𝑘1subscript𝑦𝑖𝑠subscript𝑤𝑖𝑗𝑟\displaystyle W^{\alpha_{1}\dots\alpha_{L}}_{\beta_{1}\dots\beta_{L}}\sim\hat{Z}\cdot\prod_{i,j=1}^{L}\prod_{k,r,s=1}^{\infty}\frac{(1-\widehat{Q}_{i,j}\,Q_{\rho}^{k-1}x_{i,r}y_{i-j,s})(1-\widehat{Q}^{v}_{i,j}Q_{\rho}^{k-1}z_{i,s}w_{i+j-1,r})}{(1-\overline{Q}_{i,j}Q_{\rho}^{k-1}x_{i,r}z_{i-j,s})(1-\dot{Q}_{i,j}Q_{\rho}^{k-1}y_{i,s}w_{i+j,r})}\,, (IV.7)

where for brevity we omitted an overall prefactor independent of the external legs, which will be fixed by the normalisation in the end. The modular parameter is given by

Qρ=Q~1Q~2Q~Lsubscript𝑄𝜌subscript~𝑄1subscript~𝑄2subscript~𝑄𝐿\displaystyle Q_{\rho}=\widetilde{Q}_{1}\widetilde{Q}_{2}\dots\widetilde{Q}_{L} with Q~i=QhiQvi.subscript~𝑄𝑖subscript𝑄subscript𝑖subscript𝑄subscript𝑣𝑖\displaystyle\widetilde{Q}_{i}=Q_{h_{i}}Q_{v_{i}}\,. (IV.8)

In (IV.7), the arguments xisubscript𝑥𝑖x_{i}, yisubscript𝑦𝑖y_{i}, wisubscript𝑤𝑖w_{i} and zisubscript𝑧𝑖z_{i} are given in Eq.(IV.4) and we have introduced the following shorthand notation

Q^i,j=Qhik=1j1Q~ik,Q^i,jv=Qvik=1j1Q~i+kformulae-sequencesubscript^𝑄𝑖𝑗subscript𝑄subscript𝑖superscriptsubscriptproduct𝑘1𝑗1subscript~𝑄𝑖𝑘superscriptsubscript^𝑄𝑖𝑗𝑣subscript𝑄subscript𝑣𝑖superscriptsubscriptproduct𝑘1𝑗1subscript~𝑄𝑖𝑘\displaystyle\widehat{Q}_{i,j}=Q_{h_{i}}\prod_{k=1}^{j-1}\widetilde{Q}_{i-k}\,,\hskip 14.22636pt\widehat{Q}_{i,j}^{v}=Q_{v_{i}}\prod_{k=1}^{j-1}\widetilde{Q}_{i+k}
Q¯i,j={1if j=LQhiQvijk=1j1Q~ikif jLsubscript¯𝑄𝑖𝑗cases1if 𝑗𝐿subscript𝑄subscript𝑖subscript𝑄subscript𝑣𝑖𝑗superscriptsubscriptproduct𝑘1𝑗1subscript~𝑄𝑖𝑘if 𝑗𝐿\displaystyle\overline{Q}_{i,j}=\begin{cases}1&\quad\text{if }j=L\\ Q_{h_{i}}Q_{v_{i-j}}\prod_{k=1}^{j-1}\widetilde{Q}_{i-k}&\quad\text{if }j\neq L\\ \end{cases}
Q˙i,j=Q~i+1Q~i+j.subscript˙𝑄𝑖𝑗subscript~𝑄𝑖1subscript~𝑄𝑖𝑗\displaystyle\dot{Q}_{i,j}=\widetilde{Q}_{i+1}\dots\widetilde{Q}_{i+j}\,. (IV.9)

Finally, upon reinstating the appropriate normalisation factor WL()subscript𝑊𝐿W_{L}(\emptyset) (which is interpreted as the closed string amplitude and is defined below), Eq. (IV.7) can be written as follows (see Eq. (A.2) for the definition of functions 𝒥αβsubscript𝒥𝛼𝛽\mathcal{J}_{\alpha\beta}),

Wβ1βLα1αL=WL()Z^i,j=1L𝒥αiβj(Q^i,ij;q,t)𝒥βjαi((Q^i,ij)1Qρ;q,t)𝒥αiαj(Q¯i,ijq/t;q,t)𝒥βjβi(Q˙i,jit/q;q,t),subscriptsuperscript𝑊subscript𝛼1subscript𝛼𝐿subscript𝛽1subscript𝛽𝐿subscript𝑊𝐿^𝑍superscriptsubscriptproduct𝑖𝑗1𝐿subscript𝒥subscript𝛼𝑖subscript𝛽𝑗subscript^𝑄𝑖𝑖𝑗𝑞𝑡subscript𝒥subscript𝛽𝑗subscript𝛼𝑖superscriptsubscript^𝑄𝑖𝑖𝑗1subscript𝑄𝜌𝑞𝑡subscript𝒥subscript𝛼𝑖subscript𝛼𝑗subscript¯𝑄𝑖𝑖𝑗𝑞𝑡𝑞𝑡subscript𝒥subscript𝛽𝑗subscript𝛽𝑖subscript˙𝑄𝑖𝑗𝑖𝑡𝑞𝑞𝑡\displaystyle W^{\alpha_{1}\dots\alpha_{L}}_{\beta_{1}\dots\beta_{L}}=W_{L}(\emptyset)\cdot\hat{Z}\cdot\prod_{i,j=1}^{L}\frac{\mathcal{J}_{\alpha_{i}\beta_{j}}(\widehat{Q}_{i,i-j};q,t)\mathcal{J}_{\beta_{j}\alpha_{i}}((\widehat{Q}_{i,i-j})^{-1}Q_{\rho};q,t)}{\mathcal{J}_{\alpha_{i}\alpha_{j}}(\overline{Q}_{i,i-j}\sqrt{q/t};q,t)\mathcal{J}_{\beta_{j}\beta_{i}}(\dot{Q}_{i,j-i}\sqrt{t/q};q,t)}\,, (IV.10)

where

WL()=i,j=1Lk,r,s=1(1Q^i,jQρk1qr12ts12)(1Q^i,j1Qρkqs12tr12)(1Q¯i,jQρk1qrts1)(1Q˙i,jQρk1qs1tr).subscript𝑊𝐿superscriptsubscriptproduct𝑖𝑗1𝐿superscriptsubscriptproduct𝑘𝑟𝑠11subscript^𝑄𝑖𝑗superscriptsubscript𝑄𝜌𝑘1superscript𝑞𝑟12superscript𝑡𝑠121subscriptsuperscript^𝑄1𝑖𝑗superscriptsubscript𝑄𝜌𝑘superscript𝑞𝑠12superscript𝑡𝑟121subscript¯𝑄𝑖𝑗superscriptsubscript𝑄𝜌𝑘1superscript𝑞𝑟superscript𝑡𝑠11subscript˙𝑄𝑖𝑗superscriptsubscript𝑄𝜌𝑘1superscript𝑞𝑠1superscript𝑡𝑟\displaystyle W_{L}(\emptyset)=\prod_{i,j=1}^{L}\prod_{k,r,s=1}^{\infty}\frac{(1-\widehat{Q}_{i,j}\,Q_{\rho}^{k-1}q^{r-\frac{1}{2}}t^{s-\frac{1}{2}})(1-\widehat{Q}^{-1}_{i,j}Q_{\rho}^{k}q^{s-\frac{1}{2}}t^{r-\frac{1}{2}})}{(1-\overline{Q}_{i,j}Q_{\rho}^{k-1}q^{r}t^{s-1})(1-\dot{Q}_{i,j}Q_{\rho}^{k-1}q^{s-1}t^{r})}\,. (IV.11)

While the numerator of (IV.10) can in principle be further simplified by combining the 𝒥αβ𝒥βαsubscript𝒥𝛼𝛽subscript𝒥𝛽𝛼\mathcal{J}_{\alpha\beta}\mathcal{J}_{\beta\alpha} into ϑαβsubscriptitalic-ϑ𝛼𝛽\vartheta_{\alpha\beta} following (A.5), a similar simplification for the denominator requires a gluing of the external legs. We therefore postpone these steps to the next section, where the latter is considered in detail.

There is a more intuitive way of understanding the structure of (IV.10). The open string amplitude in Eq. (IV.7) can be understood in terms of counting holomorphic curves in the presence of Lagrangian branes Iqbal:2004ne . Recall that, if the external legs of the strip diagram are not glued (as in Fig. 7), there are only two types of curves:

  1. (i)

    curves with local geometry 𝒪(1,1)1maps-to𝒪11superscript1{\cal O}(-1,-1)\mapsto\mathbb{P}^{1}

  2. (ii)

    curves with local geometry 𝒪(2, 0)1maps-to𝒪2 0superscript1{\cal O}(-2,\ \ 0)\mapsto\mathbb{P}^{1}

If we place Lagrangian branes Aganagic:2000gs on (pairs of) external legs of the web diagram, the curves contributing to the open string amplitude (IV.7) are of type (i) if the external legs are on different sides of the diagram and of type (ii) if they are on the same side, respectively. In the former case, for two external legs labelled by αisubscript𝛼𝑖\alpha_{i} and βjsubscript𝛽𝑗\beta_{j}, these curves are shown in Fig. 8(a) and Fig. 8(b), respectively, and their contributions to the open string amplitude Wβ1βLα1αLsubscriptsuperscript𝑊subscript𝛼1subscript𝛼𝐿subscript𝛽1subscript𝛽𝐿W^{\alpha_{1}\dots\alpha_{L}}_{\beta_{1}\dots\beta_{L}} in (IV.7) are

Zαiβj(a)=a,b=1(1Qqaαi,bt12tbβj,a12),subscriptsuperscript𝑍𝑎subscript𝛼𝑖subscript𝛽𝑗superscriptsubscriptproduct𝑎𝑏11𝑄superscript𝑞𝑎subscriptsuperscript𝛼𝑡𝑖𝑏12superscript𝑡𝑏subscript𝛽𝑗𝑎12\displaystyle\displaystyle Z^{(a)}_{\alpha_{i}\beta_{j}}=\prod_{a,b=1}^{\infty}(1-Q\,q^{a-\alpha^{t}_{i,b}-\frac{1}{2}}t^{b-\beta_{j,a}-\frac{1}{2}})\,,
Zβiαj(b)=a,b=1(1Qqaβi,bt12tbαj,a12).subscriptsuperscript𝑍𝑏subscript𝛽𝑖subscript𝛼𝑗superscriptsubscriptproduct𝑎𝑏11𝑄superscript𝑞𝑎subscriptsuperscript𝛽𝑡𝑖𝑏12superscript𝑡𝑏subscript𝛼𝑗𝑎12\displaystyle Z^{(b)}_{\beta_{i}\alpha_{j}}=\prod_{a,b=1}^{\infty}(1-Q\,q^{a-\beta^{t}_{i,b}-\frac{1}{2}}t^{b-\alpha_{j,a}-\frac{1}{2}})\,. (IV.12)

The curves of type (ii) with branes on external legs on the same side labelled by αi,αjsubscript𝛼𝑖subscript𝛼𝑗\alpha_{i},\alpha_{j} or βi,βjsubscript𝛽𝑖subscript𝛽𝑗\beta_{i},\beta_{j} are shown in Fig. 8(b) and contribute to the open string amplitude,

Zαiαj(c)=a,b=1(1Qqaαj,bttbαi,a1),subscriptsuperscript𝑍𝑐subscript𝛼𝑖subscript𝛼𝑗superscriptsubscriptproduct𝑎𝑏11𝑄superscript𝑞𝑎subscriptsuperscript𝛼𝑡𝑗𝑏superscript𝑡𝑏subscript𝛼𝑖𝑎1\displaystyle\displaystyle Z^{(c)}_{\alpha_{i}\alpha_{j}}=\prod_{a,b=1}^{\infty}(1-Q\,q^{a-\alpha^{t}_{j,b}}t^{b-\alpha_{i,a}-1})\,,
Zβiβj(d)=a,b=1(1Qqaβj,bt1tbβi,a).subscriptsuperscript𝑍𝑑subscript𝛽𝑖subscript𝛽𝑗superscriptsubscriptproduct𝑎𝑏11𝑄superscript𝑞𝑎subscriptsuperscript𝛽𝑡𝑗𝑏1superscript𝑡𝑏subscript𝛽𝑖𝑎\displaystyle Z^{(d)}_{\beta_{i}\beta_{j}}=\prod_{a,b=1}^{\infty}(1-Q\,q^{a-\beta^{t}_{j,b}-1}t^{b-\beta_{i,a}})\,. (IV.13)

Here, for all four cases ln(Q)𝑄-\ln(Q) is the Kähler parameter associated with the only holomorphic curve in the geometry. In order to describe the area of this curve in terms of the Kähler parameters of the strip diagram, the parameters a^1,,Lsubscript^𝑎1𝐿\widehat{a}_{1,\ldots,L} and b^1,,Lsubscript^𝑏1𝐿\widehat{b}_{1,\ldots,L} in (IV.6) are very useful (see Fig. 7). In terms of the latter, we have for (IV.9)

Q^i,j=exp(hik=1j1b^ik),Q˙i,j=exp(k=1jb^i+k),formulae-sequencesubscript^𝑄𝑖𝑗expsubscript𝑖superscriptsubscript𝑘1𝑗1subscript^𝑏𝑖𝑘subscript˙𝑄𝑖𝑗superscriptsubscript𝑘1𝑗subscript^𝑏𝑖𝑘\displaystyle\widehat{Q}_{i,j}=\text{exp}(-h_{i}-\sum_{k=1}^{j-1}\widehat{b}_{i-k})\,,\hskip 8.5359pt\dot{Q}_{i,j}=\exp(-\sum_{k=1}^{j}\widehat{b}_{i+k})\,,
Q¯i,j={1if j=L,exp(k=1ja^ik)if jL.subscript¯𝑄𝑖𝑗cases1if 𝑗𝐿superscriptsubscript𝑘1𝑗subscript^𝑎𝑖𝑘if 𝑗𝐿\displaystyle\overline{Q}_{i,j}=\begin{cases}1&\quad\text{if }j=L\,,\\ \exp(-\sum_{k=1}^{j}\widehat{a}_{i-k})&\quad\text{if }j\neq L\,.\\ \end{cases} (IV.14)

Note that gluing the external legs (as e.g. in Fig. 9) of the strip allows infinitely many holomorphic curve between any two external lines by going around the strip a number of times before ending. As there are no other holomorphic curves, the open string amplitude associated with the glued strip is an infinite product over the “winding” in addition to a product over distinct pairs of external legs. However, each pair of legs still gives rise to factors of the type shown in Eq. (IV.12) and Eq. (IV.13).

βisubscript𝛽𝑖\beta_{i}αjsubscript𝛼𝑗\alpha_{j}βisubscript𝛽𝑖\beta_{i}αjsubscript𝛼𝑗\alpha_{j}αisubscript𝛼𝑖\alpha_{i}αjsubscript𝛼𝑗\alpha_{j}βi𝛽𝑖\beta{i}βjsubscript𝛽𝑗\beta_{j}(a)𝑎(a)(b)𝑏(b)(c)𝑐(c)(d)𝑑(d)
Figure 8: (a) Lagrangian branes (respresented as dashed red lines) on the resolved conifold and (b) after flop transition. (c) Lagrangian branes on 𝒪(2)𝒪(0)1maps-todirect-sum𝒪2𝒪0superscript1{\cal O}(-2)\oplus{\cal O}(0)\mapsto\mathbb{P}^{1} and (d) Lagrangian branes on 𝒪(0)𝒪(2)1maps-todirect-sum𝒪0𝒪2superscript1{\cal O}(0)\oplus{\cal O}(-2)\mapsto\mathbb{P}^{1}.

IV.3 Twisted Strip and Duality for gcd(N,M)=1gcd𝑁𝑀1\text{gcd}(N,M)=1

Having computed the general building block in (IV.10), arbitrary partition functions of the type 𝒵N,Msubscript𝒵𝑁𝑀\mathcal{Z}_{N,M} can be computed by gluing several of the W𝑊W together along the external lines. Depending on the choice of gluing parameters (and the orientation of the fundamental building block), we can obtain various different series expansions of 𝒵N,Msubscript𝒵𝑁𝑀\mathcal{Z}_{N,M}. Indeed, for the diagonal expansion, the gluing of several strips is indicated in Fig. 6(b).

As a next step, we can use (IV.10) to verify explicitly (II.1), thus generalising our checks of (III.1) to more generic configurations. As the explicit computations are rather involved, we limit ourselves to cases with gcd(N,M)=1gcd𝑁𝑀1\text{gcd}(N,M)=1, in which the shifted web (shown in Fig. 6(b)) takes the form of a single ‘staircase’ strip of length L=NM𝐿𝑁𝑀L=NM, whose external lines are glued with a shift δ𝛿\delta given by eq. (IV.1). This configuration is schematically shown in Fig. 9. As was explained in Hohenegger:2016yuv , the duality XN,MXNM,1similar-tosubscript𝑋𝑁𝑀subscript𝑋𝑁𝑀1X_{N,M}\sim X_{NM,1} (assuming gcd(N,M)=1gcd𝑁𝑀1\text{gcd}(N,M)=1) relies on iteratively using flop and symmetry transformations to change the twisted strip with shift δ>0𝛿0\delta>0 into a twisted strip with shift δ=0𝛿0\delta=0. While this procedure changes the individual intervals in the (twisted) web-diagram, it is expected that the partition function 𝒵N,Msubscript𝒵𝑁𝑀\mathcal{Z}_{N,M} remains invariant. In the following, we will show explicitly that the partition function is invariant under the transformations that changes δδ+1𝛿𝛿1\delta\longrightarrow\delta+1, which (by induction) explicitly proves (II.1) for gcd(N,M)=1gcd𝑁𝑀1\text{gcd}(N,M)=1.

m1,α1subscript𝑚1subscript𝛼1m_{1},\alpha_{1}m2,α2subscript𝑚2subscript𝛼2m_{2},\alpha_{2}m3,α3subscript𝑚3subscript𝛼3m_{3},\alpha_{3}αLδ+1tsubscriptsuperscript𝛼𝑡𝐿𝛿1\alpha^{t}_{L-\delta+1}αLδ+2tsubscriptsuperscript𝛼𝑡𝐿𝛿2\alpha^{t}_{L-\delta+2}a𝑎ah1subscript1h_{1}h2subscript2h_{2}h3subscript3h_{3}v1subscript𝑣1v_{1}v2subscript𝑣2v_{2}\cdotsα1tsubscriptsuperscript𝛼𝑡1\alpha^{t}_{1}α2tsubscriptsuperscript𝛼𝑡2\alpha^{t}_{2}vδ+1subscript𝑣𝛿1v_{\delta+1}hδ+1subscript𝛿1h_{\delta+1}mδ+1,αδ+1subscript𝑚𝛿1subscript𝛼𝛿1m_{\delta+1},\alpha_{\delta+1}\cdotsmL,αLsubscript𝑚𝐿subscript𝛼𝐿m_{L},\alpha_{L}αLδtsubscriptsuperscript𝛼𝑡𝐿𝛿\alpha^{t}_{L-\delta}vLsubscript𝑣𝐿v_{L}a𝑎aa^1subscript^𝑎1\widehat{a}_{1}a^2subscript^𝑎2\widehat{a}_{2}\cdotsa^δ+1subscript^𝑎𝛿1\widehat{a}_{\delta+1}\cdotsa^Lsubscript^𝑎𝐿\widehat{a}_{L}a^Lδsubscript^𝑎𝐿𝛿\widehat{a}_{L-\delta}a^Lδ+1subscript^𝑎𝐿𝛿1\widehat{a}_{L-\delta+1}\cdotsa^1subscript^𝑎1\widehat{a}_{1}a^Lδ1subscript^𝑎𝐿𝛿1\widehat{a}_{L-\delta-1}\cdots\cdotsS𝑆S\cdots\cdotsR(δ)superscript𝑅𝛿-R^{(\delta)}
Figure 9: Strip of length L𝐿L and shift δ𝛿\delta with parametrisation suitable for the topological vertex computation.

IV.3.1 Parametrisation

In this section, we consider a twisted web diagram of length L=NM𝐿𝑁𝑀L=NM in which the external (diagonal) legs are glued together (albeit with a shift δ𝛿\delta), so not all line segments in Fig. 9 are independent one another. Instead, they have to satisfy the following consistency conditions

vi+misubscript𝑣𝑖subscript𝑚𝑖\displaystyle v_{i}+m_{i} =vi+δ+1+mi+1,absentsubscript𝑣𝑖𝛿1subscript𝑚𝑖1\displaystyle=v_{i+\delta+1}+m_{i+1}\,,
hi+1+mi+1subscript𝑖1subscript𝑚𝑖1\displaystyle h_{i+1}+m_{i+1} =hi+δ+1+mi,absentsubscript𝑖𝛿1subscript𝑚𝑖\displaystyle=h_{i+\delta+1}+m_{i}\,, (IV.15)

which leave a total of L+2𝐿2L+2 independent parameters. Eliminating the misubscript𝑚𝑖m_{i} from (IV.15), we obtain

a^i=vi+hi+1=vi+δ+1+hi+δ+1,subscript^𝑎𝑖subscript𝑣𝑖subscript𝑖1subscript𝑣𝑖𝛿1subscript𝑖𝛿1\widehat{a}_{i}=v_{i}+h_{i+1}=v_{i+\delta+1}+h_{i+\delta+1}\,, (IV.16)

which in terms of the variables (IV.6) corresponds to a^i=b^i+δ+1subscript^𝑎𝑖subscript^𝑏𝑖𝛿1\widehat{a}_{i}=\widehat{b}_{i+\delta+1}, as has already been taken into account in Fig. 9. There, we have also introduced the (diagonal) distance S𝑆S between identified external legs, which is represented by the blue curve. In Fig. 9, we chose to measure this distance between the diagonal intervals carrying the integer partition function α1subscript𝛼1\alpha_{1} and α1tsuperscriptsubscript𝛼1𝑡\alpha_{1}^{t}, respectively. In fact, S𝑆S is the same for any pair of identified legs (we adopt a notation where hi=hi+Lsubscript𝑖subscript𝑖𝐿h_{i}=h_{i+L} and similarly a^i=a^i+Lsubscript^𝑎𝑖subscript^𝑎𝑖𝐿\widehat{a}_{i}=\widehat{a}_{i+L})

S=hi(Lδ1)+r=1Lδ1a^ir,     1iL,formulae-sequence𝑆subscript𝑖𝐿𝛿1superscriptsubscript𝑟1𝐿𝛿1subscript^𝑎𝑖𝑟1𝑖𝐿\displaystyle S=h_{i-(L-\delta-1)}+\sum_{r=1}^{L-\delta-1}\widehat{a}_{i-r}\,,\,\,\,\,\,1\leq i\leq L\,, (IV.17)

due to the consistency conditions (IV.15) and (IV.16). We can thus express the horizontal distances as hi=Sk=δ+2La^iksubscript𝑖𝑆superscriptsubscript𝑘𝛿2𝐿subscript^𝑎𝑖𝑘h_{i}=S-\sum_{k=\delta+2}^{L}\widehat{a}_{i-k}\, which, for the shorthand notation in (IV.14), implies that

Q^i,j={exp(S+r=j+δ+1La^ir)if j+δLexp(Sr=1j+δa^irif j+δ>L\displaystyle\widehat{Q}_{i,j}=\begin{cases}\exp(-S+\sum_{r=j+\delta+1}^{L}\widehat{a}_{i-r})&\text{if }j+\delta\leq L\\[4.0pt] \exp(-S-\sum_{r=1}^{j+\delta}\widehat{a}_{i-r}&\text{if }j+\delta>L\\ \end{cases}

With this notation, the partition function associated with the twisted diagram in Fig. 9 can be written as

𝒵L=NM,1(δ)={α}(i=1LQmi|αi|)WαLδ+1αLδα1αL(a^i,S,ϵ1,ϵ2).superscriptsubscript𝒵𝐿𝑁𝑀1𝛿subscript𝛼superscriptsubscriptproduct𝑖1𝐿superscriptsubscript𝑄subscript𝑚𝑖subscript𝛼𝑖subscriptsuperscript𝑊subscript𝛼1subscript𝛼𝐿subscript𝛼𝐿𝛿1subscript𝛼𝐿𝛿subscript^𝑎𝑖𝑆subscriptitalic-ϵ1subscriptitalic-ϵ2\displaystyle\mathcal{Z}_{L=NM,1}^{(\delta)}=\sum_{\{\alpha\}}\left(\prod_{i=1}^{L}Q_{m_{i}}^{|\alpha_{i}|}\right)\,W^{\alpha_{1}\dots\alpha_{L}}_{\alpha_{L-\delta+1}\dots\alpha_{L-\delta}}(\widehat{a}_{i},S,\epsilon_{1},\epsilon_{2})\,. (IV.18)

Here, we have added the explicit superscript (δ)𝛿(\delta) to indicate the partition function associated with the twisted web diagram with shift δ𝛿\delta. We will show in the next section that δ𝛿\delta in (IV.18) (for gcd(N,M)=1gcd𝑁𝑀1\text{gcd}(N,M)=1) can be arbitrarily shifted, provided that we apply the flop and symmetry transformation reviewed in appendix C to the Kähler parameters. Therefore 𝒵L=NM,1(δ)superscriptsubscript𝒵𝐿𝑁𝑀1𝛿\mathcal{Z}_{L=NM,1}^{(\delta)} is in fact identical to 𝒵N,M(ω,ϵ1,2)subscript𝒵𝑁𝑀𝜔subscriptitalic-ϵ12\mathcal{Z}_{N,M}(\omega,\epsilon_{1,2}) up to an appropriate transformation of ω𝜔\omega. It was proposed in Hohenegger:2016yuv (and checked at a particular region in the moduli space) that this also holds for gcd(N,M)>1gcd𝑁𝑀1\text{gcd}(N,M)>1.

Furthermore, W𝑊W in (IV.18) follows from the generic expression (IV.10) by identifying the partitions

βi=αi+Lδ1,subscript𝛽𝑖subscript𝛼𝑖𝐿𝛿1\displaystyle\beta_{i}=\alpha_{i+L-\delta-1}\,, i=1,,Lfor-all𝑖1𝐿\displaystyle\forall i=1,\ldots,L (IV.19)

where we recall that the latter are cyclically identified, i.e. αi=αi+Lsubscript𝛼𝑖subscript𝛼𝑖𝐿\alpha_{i}=\alpha_{i+L}). Concretely, we find

WαLδ+1αLδα1αL(a^i,S,ϵ1,ϵ2)=WL()×[(tq)L12QSLQρLδ1]|α1|++|αL|×i,j=1Lϑαiαj(Q^i,ijδ;ρ)ϑαiαj(Q¯i,ijq/t;ρ)subscriptsuperscript𝑊subscript𝛼1subscript𝛼𝐿subscript𝛼𝐿𝛿1subscript𝛼𝐿𝛿subscript^𝑎𝑖𝑆subscriptitalic-ϵ1subscriptitalic-ϵ2subscript𝑊𝐿superscriptdelimited-[]superscript𝑡𝑞𝐿12superscriptsubscript𝑄𝑆𝐿superscriptsubscript𝑄𝜌𝐿𝛿1subscript𝛼1subscript𝛼𝐿superscriptsubscriptproduct𝑖𝑗1𝐿subscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑗subscript^𝑄𝑖𝑖𝑗𝛿𝜌subscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑗subscript¯𝑄𝑖𝑖𝑗𝑞𝑡𝜌\displaystyle W^{\alpha_{1}\dots\alpha_{L}}_{\alpha_{L-\delta+1}\dots\alpha_{L-\delta}}(\widehat{a}_{i},S,\epsilon_{1},\epsilon_{2})=W_{L}(\emptyset)\times\Big{[}\Big{(}\frac{t}{q}\Big{)}^{\frac{L-1}{2}}\frac{Q_{S}^{L}}{Q_{\rho}^{L-\delta-1}}\Big{]}^{|\alpha_{1}|+\dots+|\alpha_{L}|}\times\prod_{i,j=1}^{L}\frac{\vartheta_{\alpha_{i}\alpha_{j}}(\widehat{Q}_{i,i-j-\delta};\rho)}{\vartheta_{\alpha_{i}\alpha_{j}}(\overline{Q}_{i,i-j}\sqrt{q/t};\rho)} (IV.20)

Here, we introduced

QS=eS,subscript𝑄𝑆superscript𝑒𝑆\displaystyle Q_{S}=e^{-S}\,, and Qρ=e2πiρ,subscript𝑄𝜌superscript𝑒2𝜋𝑖𝜌\displaystyle Q_{\rho}=e^{2\pi i\rho}\,, (IV.21)

with ρ=i2πi=1La^i𝜌𝑖2𝜋superscriptsubscript𝑖1𝐿subscript^𝑎𝑖\rho=\frac{i}{2\pi}\sum_{i=1}^{L}\widehat{a}_{i}. Furthermore, we have used the identities (A.5) and (A.6) to combine the 𝒥αβsubscript𝒥𝛼𝛽\mathcal{J}_{\alpha\beta} in (IV.10) into ϑitalic-ϑ\vartheta-functions (cancelling the factor Z^^𝑍\hat{Z} in the process). The full partition function (IV.18) also depends on the parameters misubscript𝑚𝑖m_{i}, which are not all independent. Indeed, by rewriting (IV.15) in terms of (independent) parameters S𝑆S and a^isubscript^𝑎𝑖\widehat{a}_{i}, we get the following recursive relation for misubscript𝑚𝑖m_{i}

mi+1=mi+k=δ+1L1α^ikk=1Lδ1α^ik.subscript𝑚𝑖1subscript𝑚𝑖superscriptsubscript𝑘𝛿1𝐿1subscript^𝛼𝑖𝑘superscriptsubscript𝑘1𝐿𝛿1subscript^𝛼𝑖𝑘\displaystyle m_{i+1}=m_{i}+\sum_{k=\delta+1}^{L-1}\widehat{\alpha}_{i-k}-\sum_{k=1}^{L-\delta-1}\widehat{\alpha}_{i-k}\,. (IV.22)

This implies that only one of the L𝐿L many parameters misubscript𝑚𝑖m_{i} can be chosen to be independent. This freedom is parametrised by R(δ)superscript𝑅𝛿R^{(\delta)}: in Fig. 9, it is shown as the vertical distance between the external legs labelled by the partition α1subscript𝛼1\alpha_{1} and α1tsuperscriptsubscript𝛼1𝑡\alpha_{1}^{t}. Similar to the parameter S𝑆S due to the consistency conditions (IV.15) (and equivalently (IV.22)), this length is the same for any pair of partitions (αi,αit)subscript𝛼𝑖superscriptsubscript𝛼𝑖𝑡(\alpha_{i},\alpha_{i}^{t})

R(δ)=mik=1Lδ1vik,superscript𝑅𝛿subscript𝑚𝑖superscriptsubscript𝑘1𝐿𝛿1subscript𝑣𝑖𝑘\displaystyle R^{(\delta)}=m_{i}-\sum_{k=1}^{L-\delta-1}v_{i-k}\,, i=1,,L=NM.formulae-sequencefor-all𝑖1𝐿𝑁𝑀\displaystyle\forall\,i=1,\ldots,L=NM\,.

We remark that (R(δ),S,a^1,,L)superscript𝑅𝛿𝑆subscript^𝑎1𝐿(R^{(\delta)},S,\widehat{a}_{1,\ldots,L}) are L+2=NM+2𝐿2𝑁𝑀2L+2=NM+2 independent variables and therefore describe a maximally independent set of Kähler parameters for XN,Msubscript𝑋𝑁𝑀X_{N,M}.

IV.3.2 Flop Transformations

Our strategy for proving (II.1) for gcd(N,M)=1gcd𝑁𝑀1\text{gcd}(N,M)=1 is to show that

𝒵L,1(δ)(ω,ϵ1,ϵ2)=𝒵L,1(δ+1)(ω,ϵ1,ϵ2)superscriptsubscript𝒵𝐿1𝛿𝜔subscriptitalic-ϵ1subscriptitalic-ϵ2superscriptsubscript𝒵𝐿1𝛿1superscript𝜔subscriptitalic-ϵ1subscriptitalic-ϵ2\displaystyle\mathcal{Z}_{L,1}^{(\delta)}(\omega,\epsilon_{1},\epsilon_{2})=\mathcal{Z}_{L,1}^{(\delta+1)}(\omega^{\prime},\epsilon_{1},\epsilon_{2})\, (IV.23)

for the partition function defined in (IV.18): indeed, 𝒵L,1(δ+1)superscriptsubscript𝒵𝐿1𝛿1\mathcal{Z}_{L,1}^{(\delta+1)} is associated with a twisted web diagram (with shift δ+1𝛿1\delta+1) that can be related to Fig. 9 through a series of flop and symmetry transformations, that also relate the Kähler parameters ω𝜔\omega and ωsuperscript𝜔\omega^{\prime}. This duality was first discussed in detail in Hohenegger:2016yuv and is reviewed in appendix C.

The parameters (IV.16) and (IV.17) in the transformed diagram are the same as those defined in the original diagram

a^isuperscriptsubscript^𝑎𝑖\displaystyle\widehat{a}_{i}^{\prime} =vi+hi+1=hi+hi+vi+hi+1=a^iabsentsuperscriptsubscript𝑣𝑖superscriptsubscript𝑖1subscript𝑖subscript𝑖subscript𝑣𝑖subscript𝑖1subscript^𝑎𝑖\displaystyle=v_{i}^{\prime}+h_{i+1}^{\prime}=-h_{i}+h_{i}+v_{i}+h_{i+1}=\widehat{a}_{i}
Ssuperscript𝑆\displaystyle S^{\prime} =hi(Lδ1)+a^i(Lδ1)+r=1Lδ2a^ir=S.absentsubscript𝑖𝐿𝛿1subscript^𝑎𝑖𝐿𝛿1superscriptsubscript𝑟1𝐿𝛿2subscript^𝑎𝑖𝑟𝑆\displaystyle=h_{i-(L-\delta-1)}+\widehat{a}_{i-(L-\delta-1)}+\sum_{r=1}^{L-\delta-2}\widehat{a}_{i-r}=S\,.

Hence, the a^isubscript^𝑎𝑖\widehat{a}_{i} always parametrise the distance between two adjacent legs and S𝑆S measures the vertical distance between two identified legs.

In order to show that (IV.23) holds, we first rewrite the building block (IV.20) in a way that makes the δ𝛿\delta dependence explicit

WαLδ+1αLδα1αL=WL()×[(tq)L12QSLQρ1K=C]|α1|++|αL|(i,j=1L1ϑαiαj(Q¯i,ijq/t;ρ))subscriptsuperscript𝑊subscript𝛼1subscript𝛼𝐿subscript𝛼𝐿𝛿1subscript𝛼𝐿𝛿subscript𝑊𝐿superscriptdelimited-[]superscript𝑡𝑞𝐿12superscriptsubscript𝑄𝑆𝐿superscriptsuperscriptsubscript𝑄𝜌1𝐾absent𝐶subscript𝛼1subscript𝛼𝐿superscriptsubscriptproduct𝑖𝑗1𝐿1subscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑗subscript¯𝑄𝑖𝑖𝑗𝑞𝑡𝜌\displaystyle W^{\alpha_{1}\dots\alpha_{L}}_{\alpha_{L-\delta+1}\dots\alpha_{L-\delta}}=W_{L}(\emptyset)\times\Big{[}\Big{(}\frac{t}{q}\Big{)}^{\frac{L-1}{2}}Q_{S}^{L}\,\overbrace{Q_{\rho}^{1-K}}^{=C}\,\Big{]}^{|\alpha_{1}|+\dots+|\alpha_{L}|}\cdot\left(\prod_{i,j=1}^{L}\frac{1}{\vartheta_{\alpha_{i}\alpha_{j}}(\overline{Q}_{i,i-j}\sqrt{q/t};\rho)}\right)
×(ijji<Kϑαiαj(Qi,j1QS))(ijjiKϑαiαj(Qi,j1QSQρ))=A(i>jijδϑαiαj(Qj,iQS))(i>jij>δϑαiαj(Qj,iQSQρ1))=B,absentsuperscriptsubscriptproductFRACOP𝑖𝑗𝑗𝑖𝐾subscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑗superscriptsubscript𝑄𝑖𝑗1subscript𝑄𝑆subscriptproductFRACOP𝑖𝑗𝑗𝑖𝐾subscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑗superscriptsubscript𝑄𝑖𝑗1subscript𝑄𝑆subscript𝑄𝜌absent𝐴superscriptsubscriptproductFRACOP𝑖𝑗𝑖𝑗𝛿subscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑗subscript𝑄𝑗𝑖subscript𝑄𝑆subscriptproductFRACOP𝑖𝑗𝑖𝑗𝛿subscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑗subscript𝑄𝑗𝑖subscript𝑄𝑆superscriptsubscript𝑄𝜌1absent𝐵\displaystyle\hskip 14.22636pt\times\overbrace{\left(\prod_{{i\leq j}\atop{j-i<K}}\vartheta_{\alpha_{i}\alpha_{j}}(Q_{i,j}^{-1}Q_{S})\right)\left(\prod_{{i\leq j}\atop{j-i\geq K}}\vartheta_{\alpha_{i}\alpha_{j}}(Q_{i,j}^{-1}Q_{S}Q_{\rho})\right)}^{=A}\cdot\overbrace{\left(\prod_{{i>j}\atop{i-j\leq\delta}}\vartheta_{\alpha_{i}\alpha_{j}}(Q_{j,i}Q_{S})\right)\left(\prod_{{i>j}\atop{i-j>\delta}}\vartheta_{\alpha_{i}\alpha_{j}}(Q_{j,i}Q_{S}Q_{\rho}^{-1})\right)}^{=B}\,, (IV.24)

where K=Lδ𝐾𝐿𝛿K=L-\delta and the Qi,jsubscript𝑄𝑖𝑗Q_{i,j} are defined as follows

Qi,j=k=0ji1exp(α^i+k)subscript𝑄𝑖𝑗superscriptsubscriptproduct𝑘0𝑗𝑖1subscript^𝛼𝑖𝑘\displaystyle Q_{i,j}=\prod_{k=0}^{j-i-1}\exp(-\widehat{\alpha}_{i+k}) (IV.25)

The difference between WαLδ+1αLδα1αLsubscriptsuperscript𝑊subscript𝛼1subscript𝛼𝐿subscript𝛼𝐿𝛿1subscript𝛼𝐿𝛿W^{\alpha_{1}\dots\alpha_{L}}_{\alpha_{L-\delta+1}\dots\alpha_{L-\delta}} for the original diagram and WαLδαLδ1α1αLsubscriptsuperscript𝑊subscript𝛼1subscript𝛼𝐿subscript𝛼𝐿𝛿subscript𝛼𝐿𝛿1W^{\alpha_{1}\dots\alpha_{L}}_{\alpha_{L-\delta}\dots\alpha_{L-\delta-1}} for the diagram obtained by flop and symmetry transformations rests in the three terms A,B𝐴𝐵A,B and C𝐶C. Their respective counterparts in the shifted diagram, denoted by A,Bsuperscript𝐴superscript𝐵A^{\prime},B^{\prime} and Csuperscript𝐶C^{\prime}, are given by

Asuperscript𝐴\displaystyle A^{\prime} =(ijji<Kϑαiαj(QSQi,j))(ijjiKϑαiαj(QSQρQi,j)),absentsubscriptproductFRACOP𝑖𝑗𝑗𝑖superscript𝐾subscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑗subscript𝑄𝑆subscript𝑄𝑖𝑗subscriptproductFRACOP𝑖𝑗𝑗𝑖superscript𝐾subscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑗subscript𝑄𝑆subscript𝑄𝜌subscript𝑄𝑖𝑗\displaystyle=\left(\prod_{{i\leq j}\atop{j-i<K^{\prime}}}\vartheta_{\alpha_{i}\alpha_{j}}\left(\tfrac{Q_{S}}{Q_{i,j}}\right)\right)\left(\prod_{{i\leq j}\atop{j-i\geq K^{\prime}}}\vartheta_{\alpha_{i}\alpha_{j}}\left(\tfrac{Q_{S}Q_{\rho}}{Q_{i,j}}\right)\right)\,,
Bsuperscript𝐵\displaystyle B^{\prime} =(i>jijδϑαiαj(Qj,iQS))(i>jij>δϑαiαj(Qj,iQSQρ)),absentsubscriptproductFRACOP𝑖𝑗𝑖𝑗superscript𝛿subscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑗subscript𝑄𝑗𝑖subscript𝑄𝑆subscriptproductFRACOP𝑖𝑗𝑖𝑗superscript𝛿subscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑗subscript𝑄𝑗𝑖subscript𝑄𝑆subscript𝑄𝜌\displaystyle=\left(\prod_{{i>j}\atop{i-j\leq\delta^{\prime}}}\vartheta_{\alpha_{i}\alpha_{j}}(Q_{j,i}Q_{S})\right)\left(\prod_{{i>j}\atop{i-j>\delta^{\prime}}}\vartheta_{\alpha_{i}\alpha_{j}}\left(\tfrac{Q_{j,i}Q_{S}}{Q_{\rho}}\right)\right)\,,
Csuperscript𝐶\displaystyle C^{\prime} =Qρ1K,absentsuperscriptsubscript𝑄𝜌1superscript𝐾\displaystyle=Q_{\rho}^{1-K^{\prime}}, (IV.26)

where K=K1superscript𝐾𝐾1K^{\prime}=K-1 and δ=δ+1superscript𝛿𝛿1\delta^{\prime}=\delta+1. The difference between A𝐴A and Asuperscript𝐴A^{\prime} (respectively, B𝐵B and Bsuperscript𝐵B^{\prime}) lies in the arguments of those ϑitalic-ϑ\vartheta-functions for which ji=K𝑗𝑖superscript𝐾j-i=K^{\prime} (resp. ij=δ+1𝑖𝑗𝛿1i-j=\delta+1): they differ by a factor of Qρsubscript𝑄𝜌Q_{\rho}. The difference between C𝐶C and Csuperscript𝐶C^{\prime} is also a factor of Qρsubscript𝑄𝜌Q_{\rho}.

Finally, we also need to take account of the factors of Qmisubscript𝑄subscript𝑚𝑖Q_{m_{i}} that appear in the full partition function (IV.18). In the flopped diagram, these are given by (see (C.4))

Qmi={QmiQS2if δ=LQmiQS2r=δ+2LQairr=1Lδ1Qairelsesubscript𝑄subscriptsuperscript𝑚𝑖casessubscript𝑄subscript𝑚𝑖superscriptsubscript𝑄𝑆2if superscript𝛿𝐿subscript𝑄subscript𝑚𝑖superscriptsubscript𝑄𝑆2superscriptsubscriptproduct𝑟𝛿2𝐿subscript𝑄subscript𝑎𝑖𝑟superscriptsubscriptproduct𝑟1𝐿𝛿1subscript𝑄subscript𝑎𝑖𝑟elseQ_{m^{\prime}_{i}}=\left\{\begin{array}[]{ll}Q_{m_{i}}Q_{S}^{2}&\text{if }\delta^{\prime}=L\\ \tfrac{Q_{m_{i}}Q_{S}^{2}}{\prod_{r=\delta+2}^{L}Q_{a_{i-r}}\prod_{r=1}^{L-\delta-1}Q_{a_{i-r}}}&\text{else}\end{array}\right. (IV.27)

where we defined Qai=exp(a^i)subscript𝑄subscript𝑎𝑖subscript^𝑎𝑖Q_{a_{i}}=\exp(-\widehat{a}_{i}). In order to show that the equality (IV.23) holds, we now show that the difference between 𝒵L,1(δ)(ω,ϵ1,ϵ2)superscriptsubscript𝒵𝐿1𝛿𝜔subscriptitalic-ϵ1subscriptitalic-ϵ2\mathcal{Z}_{L,1}^{(\delta)}(\omega,\epsilon_{1},\epsilon_{2}) and 𝒵L,1(δ+1)(ω,ϵ1,ϵ2)superscriptsubscript𝒵𝐿1𝛿1superscript𝜔subscriptitalic-ϵ1subscriptitalic-ϵ2\mathcal{Z}_{L,1}^{(\delta+1)}(\omega^{\prime},\epsilon_{1},\epsilon_{2}) can be canceled by merely applying the shift identity (III.19) to the ϑitalic-ϑ\vartheta-functions mentioned above in (IV.26).

First, we consider the case when the twist in the external legs is δ=Lsuperscript𝛿𝐿\delta^{\prime}=L. Shifting the required ϑitalic-ϑ\vartheta-functions to regain the ϑitalic-ϑ\vartheta-structure of the δ=L1𝛿𝐿1\delta=L-1 strip gives

i=1Lϑαiαi(QSQρ)=i=1L(QS2Qρ1)|αi|ϑαiαi(QS)superscriptsubscriptproduct𝑖1𝐿subscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑖subscript𝑄𝑆subscript𝑄𝜌superscriptsubscriptproduct𝑖1𝐿superscriptsuperscriptsubscript𝑄𝑆2superscriptsubscript𝑄𝜌1subscript𝛼𝑖subscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑖subscript𝑄𝑆\displaystyle\prod_{i=1}^{L}\vartheta_{\alpha_{i}\alpha_{i}}(Q_{S}Q_{\rho})=\prod_{i=1}^{L}(Q_{S}^{-2}Q_{\rho}^{-1})^{|\alpha_{i}|}\vartheta_{\alpha_{i}\alpha_{i}}(Q_{S}) (IV.28)

The prefactors in (IV.28) resulting from the shift identity combine with (IV.26) and (IV.27) to reproduce the expression for 𝒵L,1(δ)(ω,ϵ1,ϵ2)superscriptsubscript𝒵𝐿1𝛿𝜔subscriptitalic-ϵ1subscriptitalic-ϵ2\mathcal{Z}_{L,1}^{(\delta)}(\omega,\epsilon_{1},\epsilon_{2}), thus proving that (IV.23) holds for δ=L1𝛿𝐿1\delta=L-1.

The computations when δLsuperscript𝛿𝐿\delta^{\prime}\neq L are more involved, so we will simply sketch them. Below we present the ϑitalic-ϑ\vartheta-functions from WαLδαLδ1α1αLsubscriptsuperscript𝑊subscript𝛼1subscript𝛼𝐿subscript𝛼𝐿𝛿subscript𝛼𝐿𝛿1W^{\alpha_{1}\dots\alpha_{L}}_{\alpha_{L-\delta}\dots\alpha_{L-\delta-1}} that need to be shifted in order to regain WαLδ+1αLδα1αLsubscriptsuperscript𝑊subscript𝛼1subscript𝛼𝐿subscript𝛼𝐿𝛿1subscript𝛼𝐿𝛿W^{\alpha_{1}\dots\alpha_{L}}_{\alpha_{L-\delta+1}\dots\alpha_{L-\delta}}. We need to distinguish different cases depending on the partition αisubscript𝛼𝑖\alpha_{i} in question. For the sake of clarity, we focus only on terms resulting from shifts that come to the power |αi|subscript𝛼𝑖|\alpha_{i}| in each separate case.

  1. 1.

    For imin(K,δ)𝑖superscript𝐾superscript𝛿i\leq\min(K^{\prime},\delta^{\prime}), we shift the following ϑitalic-ϑ\vartheta-functions

    ϑαi+δ+1αi(Qi,i+δ+1QS)ϑαiαi+K(QSQρQi,i+K)subscriptitalic-ϑsubscript𝛼𝑖𝛿1subscript𝛼𝑖subscript𝑄𝑖𝑖𝛿1subscript𝑄𝑆subscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑖superscript𝐾subscript𝑄𝑆subscript𝑄𝜌subscript𝑄𝑖𝑖superscript𝐾\displaystyle\vartheta_{\alpha_{i+\delta+1}\alpha_{i}}(Q_{i,i+\delta+1}Q_{S})\vartheta_{\alpha_{i}\alpha_{i+K^{\prime}}}\left(\frac{Q_{S}Q_{\rho}}{Q_{i,i+K^{\prime}}}\right)
    (QS2Qi,i+δ+11Qi,i+K)|αi|ϑαi+δ+1αi(Qi,i+δ+1QSQρ)similar-toabsentsuperscriptsuperscriptsubscript𝑄𝑆2superscriptsubscript𝑄𝑖𝑖𝛿11subscript𝑄𝑖𝑖superscript𝐾subscript𝛼𝑖subscriptitalic-ϑsubscript𝛼𝑖𝛿1subscript𝛼𝑖subscript𝑄𝑖𝑖𝛿1subscript𝑄𝑆subscript𝑄𝜌\displaystyle\sim\,(Q_{S}^{-2}Q_{i,i+\delta+1}^{-1}Q_{i,i+K^{\prime}})^{|\alpha_{i}|}\,\vartheta_{\alpha_{i+\delta+1}\alpha_{i}}\left(\frac{Q_{i,i+\delta+1}Q_{S}}{Q_{\rho}}\right)
    ×ϑαiαi+K(QSQi,i+K)absentsubscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑖superscript𝐾subscript𝑄𝑆subscript𝑄𝑖𝑖superscript𝐾\displaystyle\hskip 5.69046pt\times\vartheta_{\alpha_{i}\alpha_{i+K^{\prime}}}\left(\frac{Q_{S}}{Q_{i,i+K^{\prime}}}\right) (IV.29)
  2. 2.

    For i>max(K,δ)𝑖superscript𝐾superscript𝛿i>\max(K^{\prime},\delta^{\prime}), we shift the following ϑitalic-ϑ\vartheta-functions

    ϑαiαiδ1(Qiδ1,iQS)ϑαiKαi(QSQρQiK,i)subscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑖𝛿1subscript𝑄𝑖𝛿1𝑖subscript𝑄𝑆subscriptitalic-ϑsubscript𝛼𝑖superscript𝐾subscript𝛼𝑖subscript𝑄𝑆subscript𝑄𝜌subscript𝑄𝑖superscript𝐾𝑖\displaystyle\vartheta_{\alpha_{i}\alpha_{i-\delta-1}}(Q_{i-\delta-1,i}Q_{S})\vartheta_{\alpha_{i-K^{\prime}}\alpha_{i}}\left(\frac{Q_{S}Q_{\rho}}{Q_{i-K^{\prime},i}}\right)
    (QS2Qiδ1,i1QiK,i)|αi|ϑαiαiδ1(Qiδ1,iQSQρ)similar-toabsentsuperscriptsuperscriptsubscript𝑄𝑆2superscriptsubscript𝑄𝑖𝛿1𝑖1subscript𝑄𝑖superscript𝐾𝑖subscript𝛼𝑖subscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑖𝛿1subscript𝑄𝑖𝛿1𝑖subscript𝑄𝑆subscript𝑄𝜌\displaystyle\sim\,(Q_{S}^{-2}Q_{i-\delta-1,i}^{-1}Q_{i-K^{\prime},i})^{|\alpha_{i}|}\,\vartheta_{\alpha_{i}\alpha_{i-\delta-1}}\left(\frac{Q_{i-\delta-1,i}Q_{S}}{Q_{\rho}}\right)
    ×ϑαiKαi(QSQiK,i)absentsubscriptitalic-ϑsubscript𝛼𝑖superscript𝐾subscript𝛼𝑖subscript𝑄𝑆subscript𝑄𝑖superscript𝐾𝑖\displaystyle\hskip 5.69046pt\times\vartheta_{\alpha_{i-K^{\prime}}\alpha_{i}}\left(\frac{Q_{S}}{Q_{i-K^{\prime},i}}\right) (IV.30)
  3. 3.

    For min(K,δ)<imax(K,δ)superscript𝐾superscript𝛿𝑖superscript𝐾superscript𝛿\min(K^{\prime},\delta^{\prime})<i\leq\max(K^{\prime},\delta^{\prime}), we need to distinguish between two cases:

    1. (a)

      When K>δsuperscript𝐾superscript𝛿K^{\prime}>\delta^{\prime} we shift

      ϑαi+δ+1αi(Qi,i+δ+1QS)ϑαiαiδ1(QSQiδ1,i)subscriptitalic-ϑsubscript𝛼𝑖𝛿1subscript𝛼𝑖subscript𝑄𝑖𝑖𝛿1subscript𝑄𝑆subscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑖𝛿1subscript𝑄𝑆subscript𝑄𝑖𝛿1𝑖\displaystyle\vartheta_{\alpha_{i+\delta+1}\alpha_{i}}(Q_{i,i+\delta+1}Q_{S})\vartheta_{\alpha_{i}\alpha_{i-\delta-1}}\left(Q_{S}Q_{i-\delta-1,i}\right)
      (QS2QρQiδ1,i1Qi,i+δ+11)|αi|ϑαi+δ+1αi(Qi,i+δ+1QSQρ)similar-toabsentsuperscriptsuperscriptsubscript𝑄𝑆2subscript𝑄𝜌superscriptsubscript𝑄𝑖𝛿1𝑖1superscriptsubscript𝑄𝑖𝑖𝛿11subscript𝛼𝑖subscriptitalic-ϑsubscript𝛼𝑖𝛿1subscript𝛼𝑖subscript𝑄𝑖𝑖𝛿1subscript𝑄𝑆subscript𝑄𝜌\displaystyle\sim\,(Q_{S}^{-2}Q_{\rho}Q_{i-\delta-1,i}^{-1}Q_{i,i+\delta+1}^{-1})^{|\alpha_{i}|}\,\vartheta_{\alpha_{i+\delta+1}\alpha_{i}}\left(\frac{Q_{i,i+\delta+1}Q_{S}}{Q_{\rho}}\right)
      ×ϑαiαiδ1(QSQiδ1,iQρ)absentsubscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑖𝛿1subscript𝑄𝑆subscript𝑄𝑖𝛿1𝑖subscript𝑄𝜌\displaystyle\hskip 5.69046pt\times\vartheta_{\alpha_{i}\alpha_{i-\delta-1}}\left(\frac{Q_{S}Q_{i-\delta-1,i}}{Q_{\rho}}\right) (IV.31)
    2. (b)

      For K<δsuperscript𝐾superscript𝛿K^{\prime}<\delta^{\prime}, we shift

      ϑαiαi+K(QSQρQi,i+K)ϑαiKαi(QSQρQiK,i)subscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑖superscript𝐾subscript𝑄𝑆subscript𝑄𝜌subscript𝑄𝑖𝑖superscript𝐾subscriptitalic-ϑsubscript𝛼𝑖superscript𝐾subscript𝛼𝑖subscript𝑄𝑆subscript𝑄𝜌subscript𝑄𝑖superscript𝐾𝑖\displaystyle\vartheta_{\alpha_{i}\alpha_{i+K^{\prime}}}\left(\frac{Q_{S}Q_{\rho}}{Q_{i,i+K^{\prime}}}\right)\vartheta_{\alpha_{i-K^{\prime}}\alpha_{i}}\left(\frac{Q_{S}Q_{\rho}}{Q_{i-K^{\prime},i}}\right)
      (QS2Qρ1QiK,iQi,i+K)|αi|ϑαiαi+K(QSQi,i+K)similar-toabsentsuperscriptsuperscriptsubscript𝑄𝑆2superscriptsubscript𝑄𝜌1subscript𝑄𝑖superscript𝐾𝑖subscript𝑄𝑖𝑖superscript𝐾subscript𝛼𝑖subscriptitalic-ϑsubscript𝛼𝑖subscript𝛼𝑖superscript𝐾subscript𝑄𝑆subscript𝑄𝑖𝑖superscript𝐾\displaystyle\sim\,(Q_{S}^{-2}Q_{\rho}^{-1}Q_{i-K^{\prime},i}Q_{i,i+K^{\prime}})^{|\alpha_{i}|}\,\vartheta_{\alpha_{i}\alpha_{i+K^{\prime}}}\left(\frac{Q_{S}}{Q_{i,i+K^{\prime}}}\right)
      ×ϑαiKαi(QSQiK,i)absentsubscriptitalic-ϑsubscript𝛼𝑖superscript𝐾subscript𝛼𝑖subscript𝑄𝑆subscript𝑄𝑖superscript𝐾𝑖\displaystyle\hskip 5.69046pt\times\vartheta_{\alpha_{i-K^{\prime}}\alpha_{i}}\left(\frac{Q_{S}}{Q_{i-K^{\prime},i}}\right) (IV.32)

In each case, the factors resulting from shifting the ϑitalic-ϑ\vartheta-functions combine with (IV.26) and (IV.27) to reproduce the expression for 𝒵L,1(δ)(ω,ϵ1,ϵ2)superscriptsubscript𝒵𝐿1𝛿𝜔subscriptitalic-ϵ1subscriptitalic-ϵ2\mathcal{Z}_{L,1}^{(\delta)}(\omega,\epsilon_{1},\epsilon_{2}), thus showing that (IV.23) holds for a generic off-set δ𝛿\delta with ω𝜔\omega and ωsuperscript𝜔\omega^{\prime} related trough the duality transformations (C.4). As the relation (IV.23) can be applied iteratively to the point δ=L𝛿𝐿\delta=L (for which 𝒵L,1(δ=L)=𝒵L,1superscriptsubscript𝒵𝐿1𝛿𝐿subscript𝒵𝐿1\mathcal{Z}_{L,1}^{(\delta=L)}=\mathcal{Z}_{L,1}), this further implies (for gcd(N,M)=1𝑁𝑀1\gcd(N,M)=1)

𝒵N,M(𝐡,𝐯,𝐦,ϵ1,2)=𝒵NM,1(𝐡,𝐯,𝐦,ϵ1,2),subscript𝒵𝑁𝑀𝐡𝐯𝐦subscriptitalic-ϵ12subscript𝒵𝑁𝑀1superscript𝐡superscript𝐯superscript𝐦subscriptitalic-ϵ12\displaystyle\mathcal{Z}_{N,M}(\mathbf{h},\mathbf{v},\mathbf{m},\epsilon_{1,2})=\mathcal{Z}_{NM,1}(\mathbf{h}^{\prime},\mathbf{v}^{\prime},\mathbf{m}^{\prime},\epsilon_{1,2})\,, (IV.33)

where the Kähler parameters (𝐡,𝐯,𝐦)𝐡𝐯𝐦(\mathbf{h},\mathbf{v},\mathbf{m}) and (𝐡,𝐯,𝐦)superscript𝐡superscript𝐯superscript𝐦(\mathbf{h}^{\prime},\mathbf{v}^{\prime},\mathbf{m}^{\prime}) are related by the duality map implied by (C.4). This proves (II.1) for gcd(N,M)=1gcd𝑁𝑀1\text{gcd}(N,M)=1. As was already argued in Hohenegger:2016yuv , this behavior is expected to hold also for gcd(N,M)>1gcd𝑁𝑀1\text{gcd}(N,M)>1, however, extending the above computations to these cases is technically more involved and will be left for forthcoming study.

V Conclusions

In this paper, we studied the topological string partition functions of double elliptically fibered Calabi-Yau threefolds XN,Msubscript𝑋𝑁𝑀X_{N,M} that give rise to a class of LSTs with 8 supercharges via F-theory compactification. We have shown by an explicit example that X3,2subscript𝑋32X_{3,2} and X6,1subscript𝑋61X_{6,1}, which are related by flop and symmetry transforms, have the same topological string partition function, hence providing an explicit proof of the duality proposed in Hohenegger:2016yuv between XN,Msubscript𝑋𝑁𝑀X_{N,M} and XNMk,ksubscript𝑋𝑁𝑀𝑘𝑘X_{\frac{NM}{k},k} (k=gcd(N,M)𝑘gcd𝑁𝑀k=\mbox{gcd}(N,M)). Indeed, while the case discussed here were characterised by gcd(N,M)=1gcd𝑁𝑀1\text{gcd}(N,M)=1 (in order to simplify the computations), we expect the duality to straightforwardly extend also to gcd(N,M)>1gcd𝑁𝑀1\text{gcd}(N,M)>1 on general grounds Li:1998hba ; Liu:2005fz ; Konishi:2006ev ; Taki:2008hb .

A logically natural question is whether the diagonal expansion of 𝒵N,Msubscript𝒵𝑁𝑀\mathcal{Z}_{N,M}, performed in this paper, can also be interpreted as an instanton expansion of a new gauge theory engineered from the XN,Msubscript𝑋𝑁𝑀X_{N,M} web. As we will discuss in paper2 , this turns out indeed the case. The parameters misubscript𝑚𝑖m_{i} can be expressed in terms of coupling constants of a quiver gauge theory related by U-dualities to the usual quiver theory on the compactified M5-branes dual to XN,Msubscript𝑋𝑁𝑀X_{N,M}. The dual gauge theories coming from the horizontal and vertical description of the XN,Msubscript𝑋𝑁𝑀X_{N,M} brane web, discussed in Hohenegger:2015btj , together with this new dual gauge theory associated with the same web gives us a ”triality” of quiver gauge theories.

The Calabi-Yau threefolds XN,Msubscript𝑋𝑁𝑀X_{N,M} are resolution of N×Msubscript𝑁subscript𝑀\mathbb{Z}_{N}\times\mathbb{Z}_{M} orbifold of X1,1subscript𝑋11X_{1,1}. Therefore, another very interesting and natural question is whether similar duality related by flop transitions can be obtained for different types of orbifolds where N×Msubscript𝑁subscript𝑀\mathbb{Z}_{N}\times\mathbb{Z}_{M} is replaced by Γ1×Γ2subscriptΓ1subscriptΓ2\Gamma_{1}\times\Gamma_{2}, with Γ1,Γ2subscriptΓ1subscriptΓ2\Gamma_{1},\Gamma_{2} some other discrete subgroups of SU(2)𝑆𝑈2SU(2).

Acknowledgement

We are grateful to Dongsu Bak, Kang-Sin Choi, Taro Kimura and Washington Taylor for useful discussions. A.I. would like to acknowledge the “2017 Simons Summer Workshop on Mathematics and Physics” for hospitality during this work. A.I. was supported in part by the Higher Education Commission grant HEC-20-2518.

Appendix A Notation and Useful Identities

In this appendix, we first introduce some of our notation and provide some useful computational identities. We start by introducing some notation concerning integer partitions. Given an integer partition λ𝜆\lambda, we denote its transpose by λtsuperscript𝜆𝑡\lambda^{t}. We define

|λ|=i=1l(λ)λi,λ2=i=1l(λ)λi2,λt2=i=1l(λt)(λit)2formulae-sequence𝜆superscriptsubscript𝑖1𝑙𝜆subscript𝜆𝑖formulae-sequencesuperscriptnorm𝜆2superscriptsubscript𝑖1𝑙𝜆superscriptsubscript𝜆𝑖2superscriptnormsuperscript𝜆𝑡2superscriptsubscript𝑖1𝑙superscript𝜆𝑡superscriptsuperscriptsubscript𝜆𝑖𝑡2\displaystyle|\lambda|=\sum_{i=1}^{l(\lambda)}\lambda_{i}\,,\quad||\lambda||^{2}=\sum_{i=1}^{l(\lambda)}\lambda_{i}^{2}\,,\quad||\lambda^{t}||^{2}=\sum_{i=1}^{l(\lambda^{t})}(\lambda_{i}^{t})^{2} (A.1)

where l(λ)𝑙𝜆l(\lambda) is the length of the partition (i.e. the number of non-zero terms). We also introduce the following functions that are indexed by two integer partitions μ𝜇\mu and ν𝜈\nu:

𝒥μν(x;t,q)=k=1Jμν(Qρk1x;t,q),subscript𝒥𝜇𝜈𝑥𝑡𝑞superscriptsubscriptproduct𝑘1subscript𝐽𝜇𝜈superscriptsubscript𝑄𝜌𝑘1𝑥𝑡𝑞\displaystyle\mathcal{J}_{\mu\nu}(x;t,q)=\prod_{k=1}^{\infty}J_{\mu\nu}(Q_{\rho}^{k-1}x;t,q)\,, (A.2)

where

Jμν(x;t,q)=subscript𝐽𝜇𝜈𝑥𝑡𝑞absent\displaystyle J_{\mu\nu}(x;t,q)= (i,j)μ(1xtνjti+12qμij+12)subscriptproduct𝑖𝑗𝜇1𝑥superscript𝑡subscriptsuperscript𝜈𝑡𝑗𝑖12superscript𝑞subscript𝜇𝑖𝑗12\displaystyle\prod_{(i,j)\in\mu}\left(1-x\,t^{\nu^{t}_{j}-i+\frac{1}{2}}q^{\mu_{i}-j+\frac{1}{2}}\right)
×\displaystyle\times (i,j)ν(1xtμjt+i12qνi+j12).subscriptproduct𝑖𝑗𝜈1𝑥superscript𝑡subscriptsuperscript𝜇𝑡𝑗𝑖12superscript𝑞subscript𝜈𝑖𝑗12\displaystyle\prod_{(i,j)\in\nu}\left(1-x\,t^{-\mu^{t}_{j}+i-\frac{1}{2}}q^{-\nu_{i}+j-\frac{1}{2}}\right)\,.

We further define

ϑμν(x;ρ)subscriptitalic-ϑ𝜇𝜈𝑥𝜌\displaystyle\vartheta_{\mu\nu}(x;\rho) =(i,j)μϑ(x1qνjt+i12tμi+j12;ρ)absentsubscriptproduct𝑖𝑗𝜇italic-ϑsuperscript𝑥1superscript𝑞superscriptsubscript𝜈𝑗𝑡𝑖12superscript𝑡subscript𝜇𝑖𝑗12𝜌\displaystyle=\prod_{(i,j)\in\mu}\vartheta\left(x^{-1}q^{-\nu_{j}^{t}+i-\frac{1}{2}}t^{-\mu_{i}+j-\frac{1}{2}};\rho\right)\,
×(i,j)νϑ(x1qμjti+12tνij+12;ρ),\displaystyle\times\prod_{(i,j)\in\nu}\vartheta\left(x^{-1}q^{\mu_{j}^{t}-i+\frac{1}{2}}t^{\nu_{i}-j+\frac{1}{2}};\rho\right)\,, (A.3)

where (with x=e2πiz𝑥superscript𝑒2𝜋𝑖𝑧x=e^{2\pi iz} and Qρ=e2πiρsubscript𝑄𝜌superscript𝑒2𝜋𝑖𝜌Q_{\rho}=e^{2\pi i\rho})

ϑ(x;ρ)italic-ϑ𝑥𝜌\displaystyle\displaystyle\vartheta(x;\rho) =\displaystyle= (x12x12)k=1(1xQρk)(1x1Qρk)superscript𝑥12superscript𝑥12superscriptsubscriptproduct𝑘11𝑥subscriptsuperscript𝑄𝑘𝜌1superscript𝑥1superscriptsubscript𝑄𝜌𝑘\displaystyle(x^{\frac{1}{2}}-x^{-\frac{1}{2}})\,\prod_{k=1}^{\infty}(1-x\,Q^{k}_{\rho})(1-x^{-1}Q_{\rho}^{k}) (A.4)
=\displaystyle= iQρ18θ1(ρ;z)k=1(1Qρk).𝑖superscriptsubscript𝑄𝜌18subscript𝜃1𝜌𝑧superscriptsubscriptproduct𝑘11superscriptsubscript𝑄𝜌𝑘\displaystyle\frac{i\,Q_{\rho}^{-\frac{1}{8}}\theta_{1}(\rho;z)}{\prod_{k=1}^{\infty}(1-Q_{\rho}^{k})}\,.

Pairs of 𝒥μνsubscript𝒥𝜇𝜈\mathcal{J}_{\mu\nu}-functions can be combined into ϑμνsubscriptitalic-ϑ𝜇𝜈\vartheta_{\mu\nu}-functions in Eq. (A.3) by utilizing the following identities

𝒥μν(x;q,t)𝒥νμ(Qρx1;q,t)subscript𝒥𝜇𝜈𝑥𝑞𝑡subscript𝒥𝜈𝜇subscript𝑄𝜌superscript𝑥1𝑞𝑡\displaystyle\mathcal{J}_{\mu\nu}(x;q,t)\mathcal{J}_{\nu\mu}(Q_{\rho}x^{-1};q,t)
=x|μ|+|ν|2qνt2μt24tμ2ν24ϑμν(x;ρ),absentsuperscript𝑥𝜇𝜈2superscript𝑞superscriptnormsuperscript𝜈𝑡2superscriptnormsuperscript𝜇𝑡24superscript𝑡superscriptnorm𝜇2superscriptnorm𝜈24subscriptitalic-ϑ𝜇𝜈𝑥𝜌\displaystyle\hskip 5.69046pt=x^{\frac{|\mu|+|\nu|}{2}}q^{\frac{||\nu^{t}||^{2}-||\mu^{t}||^{2}}{4}}t^{\frac{||\mu||^{2}-||\nu||^{2}}{4}}\,\vartheta_{\mu\nu}(x;\rho)\,, (A.5)

as well as

(1)|μ|t|μ22qμt22Z~μ(q,t)Z~μt(t,q)𝒥μμ(Qρtq;q,t)𝒥μμ(Qρqt;q,t)=1ϑμμ(qt;ρ)\displaystyle\frac{(-1)^{|\mu|}t^{|\frac{||\mu||^{2}}{2}}q^{\frac{||\mu^{t}||^{2}}{2}}\tilde{Z}_{\mu}(q,t)\tilde{Z}_{\mu^{t}}(t,q)}{\mathcal{J}_{\mu\mu}(Q_{\rho}\sqrt{\frac{t}{q}};q,t)\mathcal{J}_{\mu\mu}(Q_{\rho}\sqrt{\frac{q}{t}};q,t)}=\frac{1}{\vartheta_{\mu\mu}(\sqrt{\frac{q}{t}};\rho)}
=1ϑμμ(tq;ρ).absent1subscriptitalic-ϑ𝜇𝜇𝑡𝑞𝜌\displaystyle=\frac{1}{\vartheta_{\mu\mu}(\sqrt{\frac{t}{q}};\rho)}\,. (A.6)

In performing the calculations in Section III and Section IV, we utilized a number of computational identities. Firstly, we found the following two identities helpful for performing sums of skew Schur functions (see Macdo (page 93))

ηsηt/μ(x)sη/ν(y)subscript𝜂subscript𝑠superscript𝜂𝑡𝜇xsubscript𝑠𝜂𝜈y\displaystyle\sum_{\eta}s_{\eta^{t}/\mu}(\textbf{x})s_{\eta/\nu}(\textbf{y}) =i,j=1(1+xiyj)τsνt/τ(x)sμt/τt(y)absentsuperscriptsubscriptproduct𝑖𝑗11subscript𝑥𝑖subscript𝑦𝑗subscript𝜏subscript𝑠superscript𝜈𝑡𝜏xsubscript𝑠superscript𝜇𝑡superscript𝜏𝑡y\displaystyle=\prod_{i,j=1}^{\infty}(1+x_{i}y_{j})\sum_{\tau}s_{\nu^{t}/\tau}(\textbf{x})s_{\mu^{t}/\tau^{t}}(\textbf{y})
ηsη/μ(x)sη/ν(y)subscript𝜂subscript𝑠𝜂𝜇xsubscript𝑠𝜂𝜈y\displaystyle\sum_{\eta}s_{\eta/\mu}(\textbf{x})s_{\eta/\nu}(\textbf{y}) =i,j=1(1xiyj)1τsνt/τ(x)sμ/τ(y)absentsuperscriptsubscriptproduct𝑖𝑗1superscript1subscript𝑥𝑖subscript𝑦𝑗1subscript𝜏subscript𝑠superscript𝜈𝑡𝜏xsubscript𝑠𝜇𝜏y\displaystyle=\prod_{i,j=1}^{\infty}(1-x_{i}y_{j})^{-1}\sum_{\tau}s_{\nu^{t}/\tau}(\textbf{x})s_{\mu/\tau}(\textbf{y}) (A.7)

Secondly, we also used the following identities between products over integer partitions and infinite products Macdo

i,j=11Qqμjt+i1tνi+j1Qqi1tjsuperscriptsubscriptproduct𝑖𝑗11𝑄superscript𝑞superscriptsubscript𝜇𝑗𝑡𝑖1superscript𝑡subscript𝜈𝑖𝑗1𝑄superscript𝑞𝑖1superscript𝑡𝑗\displaystyle\prod_{i,j=1}^{\infty}\frac{1-Qq^{-\mu_{j}^{t}+i-1}t^{-\nu_{i}+j}}{1-Qq^{i-1}t^{j}} =(i,j)ν(1Qqμjt+i1tνi+j)absentsubscriptproduct𝑖𝑗𝜈1𝑄superscript𝑞superscriptsubscript𝜇𝑗𝑡𝑖1superscript𝑡subscript𝜈𝑖𝑗\displaystyle=\prod_{(i,j)\in\nu}(1-Qq^{-\mu_{j}^{t}+i-1}t^{-\nu_{i}+j})
×(i,j)μ(1Qqνjtitμij+1)\displaystyle\times\prod_{(i,j)\in\mu}(1-Qq^{\nu_{j}^{t}-i}t^{\mu_{i}-j+1})
i,j=11Qqμjt+i1tμi+j1Qqi1tjsuperscriptsubscriptproduct𝑖𝑗11𝑄superscript𝑞superscriptsubscript𝜇𝑗𝑡𝑖1superscript𝑡subscript𝜇𝑖𝑗1𝑄superscript𝑞𝑖1superscript𝑡𝑗\displaystyle\prod_{i,j=1}^{\infty}\frac{1-Qq^{-\mu_{j}^{t}+i-1}t^{-\mu_{i}+j}}{1-Qq^{i-1}t^{j}} =(i,j)μ(1Qqμjt+i1tμi+j)absentsubscriptproduct𝑖𝑗𝜇1𝑄superscript𝑞superscriptsubscript𝜇𝑗𝑡𝑖1superscript𝑡subscript𝜇𝑖𝑗\displaystyle=\prod_{(i,j)\in\mu}(1-Qq^{-\mu_{j}^{t}+i-1}t^{-\mu_{i}+j})
×(1Qqμjtitμij+1)absent1𝑄superscript𝑞superscriptsubscript𝜇𝑗𝑡𝑖superscript𝑡subscript𝜇𝑖𝑗1\displaystyle\times(1-Qq^{\mu_{j}^{t}-i}t^{\mu_{i}-j+1}) (A.8)

Finally, we also recall the following identity Taki ,

(i,j)νμjt=(i,j)μνjt.subscript𝑖𝑗𝜈superscriptsubscript𝜇𝑗𝑡subscript𝑖𝑗𝜇superscriptsubscript𝜈𝑗𝑡\sum_{(i,j)\in\nu}\mu_{j}^{t}=\sum_{(i,j)\in\mu}\nu_{j}^{t}\,. (A.9)

Appendix B Diagonal Partition Function 𝒵3,2subscript𝒵32\mathcal{Z}_{3,2}

In this appendix, we present details of the calculation of the building block W𝑊W for the computation of the diagonal expansion of 𝒵3,2subscript𝒵32\mathcal{Z}_{3,2}, which is introduced in Eq. (III.15). Using the definition of the refined topological vertex

Cλμν(t,q)=qμ22tμt22qν22Z~ν(t,q)η(qt)|η|+|λ||μ|2sλt/η(tρqν)sμ/η(qρtνt)subscript𝐶𝜆𝜇𝜈𝑡𝑞superscript𝑞superscriptnorm𝜇22superscript𝑡superscriptnormsuperscript𝜇𝑡22superscript𝑞superscriptnorm𝜈22subscript~𝑍𝜈𝑡𝑞subscript𝜂superscript𝑞𝑡𝜂𝜆𝜇2subscript𝑠superscript𝜆𝑡𝜂superscript𝑡𝜌superscript𝑞𝜈subscript𝑠𝜇𝜂superscript𝑞𝜌superscript𝑡superscript𝜈𝑡\displaystyle C_{\lambda\mu\nu}(t,q)=q^{\tfrac{||\mu||^{2}}{2}}t^{-\tfrac{||\mu^{t}||^{2}}{2}}q^{\tfrac{||\nu||^{2}}{2}}\,\tilde{Z}_{\nu}(t,q)\,\sum_{\eta}\left(\frac{q}{t}\right)^{\frac{|\eta|+|\lambda|-|\mu|}{2}}\,s_{\lambda^{t}/\eta}(t^{-\rho}q^{-\nu})\,s_{\mu/\eta}(q^{-\rho}t^{-\nu^{t}})\, (B.1)

where sμ/νsubscript𝑠𝜇𝜈s_{\mu/\nu} are skew Schur functions and

Z~ν(t,q)=(i,j)ν(1tνjti+1qνij)1,subscript~𝑍𝜈𝑡𝑞subscriptproduct𝑖𝑗𝜈superscript1superscript𝑡superscriptsubscript𝜈𝑗𝑡𝑖1superscript𝑞subscript𝜈𝑖𝑗1\displaystyle\tilde{Z}_{\nu}(t,q)=\prod_{(i,j)\in\nu}\left(1-t^{\nu_{j}^{t}-i+1}q^{\nu_{i}-j}\right)^{-1}\,, (B.2)

the expression (III.15) can be written in the form (III.16).

In what follows, we denote the whole set of variables as bold expressions, i.e. 𝐱={xi}i=1,,6𝐱subscriptsubscript𝑥𝑖𝑖16\mathbf{x}=\{x_{i}\}_{i=1,\ldots,6} and, for simplicity, we shall not consider the prefactor Z^^𝑍\hat{Z} in (III.16): indeed, the summation over the skew Schur function in (III.16) can be written in the form

G(𝕩,𝕪,𝕨,𝕫)={μ}{ν}{η}{η~}i=16(Qhi)|μi|(Qvi)|νi|sμi/η~i+3(xi+3)sμit/ηi+5(yi+5)sνit/ηi(wi)sνi/η~i+3(zi+3).𝐺𝕩𝕪𝕨𝕫subscript𝜇𝜈𝜂~𝜂superscriptsubscriptproduct𝑖16superscriptsubscript𝑄subscript𝑖subscript𝜇𝑖superscriptsubscript𝑄subscript𝑣𝑖subscript𝜈𝑖subscript𝑠subscript𝜇𝑖subscript~𝜂𝑖3subscript𝑥𝑖3subscript𝑠superscriptsubscript𝜇𝑖𝑡subscript𝜂𝑖5subscript𝑦𝑖5subscript𝑠superscriptsubscript𝜈𝑖𝑡subscript𝜂𝑖subscript𝑤𝑖subscript𝑠subscript𝜈𝑖subscript~𝜂𝑖3subscript𝑧𝑖3\displaystyle G(\mathbb{x},\mathbb{y},\mathbb{w},\mathbb{z})=\sum_{\begin{subarray}{c}\{\mu\}\{\nu\}\\ \{\eta\}\{\tilde{\eta}\}\end{subarray}}\prod_{i=1}^{6}(-Q_{h_{i}})^{|\mu_{i}|}(-Q_{v_{i}})^{|\nu_{i}|}s_{\mu_{i}/\tilde{\eta}_{i+3}}(x_{i+3})\,s_{\mu_{i}^{t}/\eta_{i+5}}(y_{i+5})s_{\nu_{i}^{t}/\eta_{i}}(w_{i})s_{\nu_{i}/\tilde{\eta}_{i+3}}(z_{i+3})\,. (B.3)

We can perform the sum over skew Schur functions using a method similar to those in Haghighat:2013gba : repeatedly using the identities (A.7) for summing skew Schur functions, we can write:

G(𝕩,𝕪,𝕨,𝕫)=P×{μ}{ν}{η}{η~}i=16𝐺𝕩𝕪𝕨𝕫𝑃subscript𝜇𝜈𝜂~𝜂superscriptsubscriptproduct𝑖16\displaystyle G(\mathbb{x},\mathbb{y},\mathbb{w},\mathbb{z})=P\times\sum_{\begin{subarray}{c}\{\mu\}\{\nu\}\\ \{\eta\}\{\tilde{\eta}\}\end{subarray}}\prod_{i=1}^{6} (Qhi)|μi|(Qvi)|νi|sμi1/η~i+2(QhiQvi1xi+3)sμi+1t/ηi(Q~iyi+5)superscriptsubscript𝑄subscript𝑖subscript𝜇𝑖superscriptsubscript𝑄subscript𝑣𝑖subscript𝜈𝑖subscript𝑠subscript𝜇𝑖1subscript~𝜂𝑖2subscript𝑄subscript𝑖subscript𝑄subscript𝑣𝑖1subscript𝑥𝑖3subscript𝑠superscriptsubscript𝜇𝑖1𝑡subscript𝜂𝑖subscript~𝑄𝑖subscript𝑦𝑖5\displaystyle(-Q_{h_{i}})^{|\mu_{i}|}(-Q_{v_{i}})^{|\nu_{i}|}s_{\mu_{i-1}/\tilde{\eta}_{i+2}}(Q_{h_{i}}Q_{v_{i-1}}x_{i+3})s_{\mu_{i+1}^{t}/\eta_{i}}(\tilde{Q}_{i}y_{i+5})
×sνi1t/ηi1(Q~iwi)sνi+1/η~i+2(Qhi+1Qvizi+3),absentsubscript𝑠superscriptsubscript𝜈𝑖1𝑡subscript𝜂𝑖1subscript~𝑄𝑖subscript𝑤𝑖subscript𝑠subscript𝜈𝑖1subscript~𝜂𝑖2subscript𝑄subscript𝑖1subscript𝑄subscript𝑣𝑖subscript𝑧𝑖3\displaystyle\times\,s_{\nu_{i-1}^{t}/\eta_{i-1}}(\tilde{Q}_{i}w_{i})s_{\nu_{i+1}/\tilde{\eta}_{i+2}}(Q_{h_{i+1}}Q_{v_{i}}z_{i+3})\,, (B.4)

where we introduced the notation Q~i=QhiQvisubscript~𝑄𝑖subscript𝑄subscript𝑖subscript𝑄subscript𝑣𝑖\tilde{Q}_{i}=Q_{h_{i}}Q_{v_{i}} and

P=i=16r,s=1𝑃superscriptsubscriptproduct𝑖16superscriptsubscriptproduct𝑟𝑠1\displaystyle P=\prod_{i=1}^{6}\prod_{r,s=1}^{\infty} (1Qhixi+3,ryi+5,s)(1Qviwi,rzi+3,s)(1QhiQvi1xi+3,rzi+2,s)(1Q~iyi+5,rwi,s)(1QhiQ~i1xi+3,ryi+4,s)(1QviQ~i+1wi+1,rzi+3,s)(1QhiQ~i1Qvi2xi+3,rzi+1,s)(1Q~iQ~i+1yi+5,rwi+1,s).1subscript𝑄subscript𝑖subscript𝑥𝑖3𝑟subscript𝑦𝑖5𝑠1subscript𝑄subscript𝑣𝑖subscript𝑤𝑖𝑟subscript𝑧𝑖3𝑠1subscript𝑄subscript𝑖subscript𝑄subscript𝑣𝑖1subscript𝑥𝑖3𝑟subscript𝑧𝑖2𝑠1subscript~𝑄𝑖subscript𝑦𝑖5𝑟subscript𝑤𝑖𝑠1subscript𝑄subscript𝑖subscript~𝑄𝑖1subscript𝑥𝑖3𝑟subscript𝑦𝑖4𝑠1subscript𝑄subscript𝑣𝑖subscript~𝑄𝑖1subscript𝑤𝑖1𝑟subscript𝑧𝑖3𝑠1subscript𝑄subscript𝑖subscript~𝑄𝑖1subscript𝑄subscript𝑣𝑖2subscript𝑥𝑖3𝑟subscript𝑧𝑖1𝑠1subscript~𝑄𝑖subscript~𝑄𝑖1subscript𝑦𝑖5𝑟subscript𝑤𝑖1𝑠\displaystyle\frac{(1-Q_{h_{i}}x_{i+3,r}y_{i+5,s})(1-Q_{v_{i}}w_{i,r}z_{i+3,s})}{(1-Q_{h_{i}}Q_{v_{i-1}}x_{i+3,r}z_{i+2,s})(1-\tilde{Q}_{i}y_{i+5,r}w_{i,s})}\,\frac{(1-Q_{h_{i}}\tilde{Q}_{i-1}x_{i+3,r}y_{i+4,s})(1-Q_{v_{i}}\tilde{Q}_{i+1}w_{i+1,r}z_{i+3,s})}{(1-Q_{h_{i}}\tilde{Q}_{i-1}Q_{v_{i-2}}x_{i+3,r}z_{i+1,s})(1-\tilde{Q}_{i}\tilde{Q}_{i+1}y_{i+5,r}w_{i+1,s})}\,.

Thus, in (B.4), we find an expression similar to the expression (B.3) except for the difference that the partitions have been shifted, e.g. the Schur functions for i=1𝑖1i=1 have been replaced in the following fashion

sμ1/η~4(x4)sμ1t/η6(y6)sν1t/η1(w1)sν1/η~4(z4)sμ6/η~3(Qh1Qv6x4)sμ2t/η1(Q~1y6)sν6t/η6(Q~1w1)sν2/η~5(Qh2Qv1z4).subscript𝑠subscript𝜇1subscript~𝜂4subscript𝑥4subscript𝑠superscriptsubscript𝜇1𝑡subscript𝜂6subscript𝑦6subscript𝑠superscriptsubscript𝜈1𝑡subscript𝜂1subscript𝑤1subscript𝑠subscript𝜈1subscript~𝜂4subscript𝑧4subscript𝑠subscript𝜇6subscript~𝜂3subscript𝑄subscript1subscript𝑄subscript𝑣6subscript𝑥4subscript𝑠superscriptsubscript𝜇2𝑡subscript𝜂1subscript~𝑄1subscript𝑦6subscript𝑠superscriptsubscript𝜈6𝑡subscript𝜂6subscript~𝑄1subscript𝑤1subscript𝑠subscript𝜈2subscript~𝜂5subscript𝑄subscript2subscript𝑄subscript𝑣1subscript𝑧4\displaystyle s_{\mu_{1}/\tilde{\eta}_{4}}(x_{4})s_{\mu_{1}^{t}/\eta_{6}}(y_{6})s_{\nu_{1}^{t}/\eta_{1}}(w_{1})s_{\nu_{1}/\tilde{\eta}_{4}}(z_{4})\to s_{\mu_{6}/\tilde{\eta}_{3}}(Q_{h_{1}}Q_{v_{6}}x_{4})s_{\mu_{2}^{t}/\eta_{1}}(\tilde{Q}_{1}y_{6})s_{\nu_{6}^{t}/\eta_{6}}(\tilde{Q}_{1}w_{1})s_{\nu_{2}/\tilde{\eta}_{5}}(Q_{h_{2}}Q_{v_{1}}z_{4})\,.

By repeating this procedure multiple times, we again obtain the quantity G𝐺G defined in (B.3), up to a prefactor P1subscript𝑃1P_{1} (The precise form of P1subscript𝑃1P_{1} turns out not to be important) and a shift of all arguments, as

G(𝕩,𝕪,𝕨,𝕫)=P1×G(Q¯𝕩,Q¯𝕪,Q¯𝕨,Q¯𝕫).𝐺𝕩𝕪𝕨𝕫subscript𝑃1𝐺¯𝑄𝕩¯𝑄𝕪¯𝑄𝕨¯𝑄𝕫\displaystyle G(\mathbb{x},\mathbb{y},\mathbb{w},\mathbb{z})=P_{1}\times G(\bar{Q}\mathbb{x},\bar{Q}\mathbb{y},\bar{Q}\mathbb{w},\bar{Q}\mathbb{z})\,. (B.5)

where we abbreviated Q¯=Q~1Q~2Q~3Q~4Q~5Q~6¯𝑄subscript~𝑄1subscript~𝑄2subscript~𝑄3subscript~𝑄4subscript~𝑄5subscript~𝑄6\bar{Q}=\tilde{Q}_{1}\tilde{Q}_{2}\tilde{Q}_{3}\tilde{Q}_{4}\tilde{Q}_{5}\tilde{Q}_{6}. Repeating further, the relation (B.5) leads to the following recursion relation for all integers n𝑛n

G(𝕩,𝕪,𝕨,𝕫)=Pn×G(Q¯n𝕩,Q¯n𝕪,Q¯n𝕨,Q¯n𝕫).𝐺𝕩𝕪𝕨𝕫subscript𝑃𝑛𝐺superscript¯𝑄𝑛𝕩superscript¯𝑄𝑛𝕪superscript¯𝑄𝑛𝕨superscript¯𝑄𝑛𝕫\displaystyle G(\mathbb{x},\mathbb{y},\mathbb{w},\mathbb{z})=P_{n}\times G(\bar{Q}^{n}\mathbb{x},\bar{Q}^{n}\mathbb{y},\bar{Q}^{n}\mathbb{w},\bar{Q}^{n}\mathbb{z})\,. (B.6)

Using the fact that limnQ¯n=0subscript𝑛superscript¯𝑄𝑛0\lim_{n\to\infty}\bar{Q}^{n}=0, we have the relation

limnsubscript𝑛\displaystyle\lim_{n\to\infty} G(Q¯n𝕩,Q¯n𝕪,Q¯n𝕨,Q¯n𝕫)𝐺superscript¯𝑄𝑛𝕩superscript¯𝑄𝑛𝕪superscript¯𝑄𝑛𝕨superscript¯𝑄𝑛𝕫\displaystyle G(\bar{Q}^{n}\mathbb{x},\bar{Q}^{n}\mathbb{y},\bar{Q}^{n}\mathbb{w},\bar{Q}^{n}\mathbb{z})
=limn{μ}{ν}{η}{η~}i6(Qhi)|μi|(Qvi)|νi|sμi/η~i+3((Q¯2)nxi+3)sμit/ηi+5(yi+5)sνit/ηi((Q¯2)nwi)sνi/η~i+3(zi+3)absentsubscript𝑛subscript𝜇𝜈𝜂~𝜂superscriptsubscriptproduct𝑖6superscriptsubscript𝑄subscript𝑖subscript𝜇𝑖superscriptsubscript𝑄subscript𝑣𝑖subscript𝜈𝑖subscript𝑠subscript𝜇𝑖subscript~𝜂𝑖3superscriptsuperscript¯𝑄2𝑛subscript𝑥𝑖3subscript𝑠superscriptsubscript𝜇𝑖𝑡subscript𝜂𝑖5subscript𝑦𝑖5subscript𝑠superscriptsubscript𝜈𝑖𝑡subscript𝜂𝑖superscriptsuperscript¯𝑄2𝑛subscript𝑤𝑖subscript𝑠subscript𝜈𝑖subscript~𝜂𝑖3subscript𝑧𝑖3\displaystyle=\lim_{n\to\infty}\sum_{\begin{subarray}{c}\{\mu\}\{\nu\}\\ \{\eta\}\{\tilde{\eta}\}\end{subarray}}\prod_{i}^{6}(-Q_{h_{i}})^{|\mu_{i}|}(-Q_{v_{i}})^{|\nu_{i}|}s_{\mu_{i}/\tilde{\eta}_{i+3}}((\bar{Q}^{2})^{n}x_{i+3})s_{\mu_{i}^{t}/\eta_{i+5}}(y_{i+5})s_{\nu_{i}^{t}/\eta_{i}}((\bar{Q}^{2})^{n}w_{i})s_{\nu_{i}/\tilde{\eta}_{i+3}}(z_{i+3})
={η}{η~}i6(Qhi)|η~i+3|(Qvi)|ηi|sη~i+3t/ηi+5(yi+5)sηit/η~i+3(zi+3),absentsubscript𝜂~𝜂superscriptsubscriptproduct𝑖6superscriptsubscript𝑄subscript𝑖subscript~𝜂𝑖3superscriptsubscript𝑄subscript𝑣𝑖subscript𝜂𝑖subscript𝑠superscriptsubscript~𝜂𝑖3𝑡subscript𝜂𝑖5subscript𝑦𝑖5subscript𝑠superscriptsubscript𝜂𝑖𝑡subscript~𝜂𝑖3subscript𝑧𝑖3\displaystyle=\sum_{\{\eta\}\{\tilde{\eta}\}}\prod_{i}^{6}(-Q_{h_{i}})^{|\tilde{\eta}_{i+3}|}(-Q_{v_{i}})^{|\eta_{i}|}s_{\tilde{\eta}_{i+3}^{t}/\eta_{i+5}}(y_{i+5})s_{\eta_{i}^{t}/\tilde{\eta}_{i+3}}(z_{i+3})\,, (B.7)

as the only non-vanishing terms in this limit correspond to μi=η~i+3subscript𝜇𝑖subscript~𝜂𝑖3\mu_{i}=\tilde{\eta}_{i+3} and νit=ηisuperscriptsubscript𝜈𝑖𝑡subscript𝜂𝑖\nu_{i}^{t}=\eta_{i}. The skew Schur functions are non-zero when ηi+2η~itsubscript𝜂𝑖2superscriptsubscript~𝜂𝑖𝑡\eta_{i+2}\subset\tilde{\eta}_{i}^{t} and η~i+3ηitsubscript~𝜂𝑖3superscriptsubscript𝜂𝑖𝑡\tilde{\eta}_{i+3}\subset\eta_{i}^{t}, such that we have

η3η~1t,η4η~2t,η5η~3t,η6η~4t,η1η~5t,η2η~6tformulae-sequencesubscript𝜂3superscriptsubscript~𝜂1𝑡formulae-sequencesubscript𝜂4superscriptsubscript~𝜂2𝑡formulae-sequencesubscript𝜂5superscriptsubscript~𝜂3𝑡formulae-sequencesubscript𝜂6superscriptsubscript~𝜂4𝑡formulae-sequencesubscript𝜂1superscriptsubscript~𝜂5𝑡subscript𝜂2superscriptsubscript~𝜂6𝑡\displaystyle\eta_{3}\subset\tilde{\eta}_{1}^{t}\,,\eta_{4}\subset\tilde{\eta}_{2}^{t}\,,\eta_{5}\subset\tilde{\eta}_{3}^{t}\,,\eta_{6}\subset\tilde{\eta}_{4}^{t}\,,\eta_{1}\subset\tilde{\eta}_{5}^{t}\,,\eta_{2}\subset\tilde{\eta}_{6}^{t}
η~4η1t,η~5η2t,η~6η3t,η~1η4t,η~2η5t,η~3η6t.formulae-sequencesubscript~𝜂4superscriptsubscript𝜂1𝑡formulae-sequencesubscript~𝜂5superscriptsubscript𝜂2𝑡formulae-sequencesubscript~𝜂6superscriptsubscript𝜂3𝑡formulae-sequencesubscript~𝜂1superscriptsubscript𝜂4𝑡formulae-sequencesubscript~𝜂2superscriptsubscript𝜂5𝑡subscript~𝜂3superscriptsubscript𝜂6𝑡\displaystyle\tilde{\eta}_{4}\subset\eta_{1}^{t}\,,\tilde{\eta}_{5}\subset\eta_{2}^{t}\,,\tilde{\eta}_{6}\subset\eta_{3}^{t}\,,\tilde{\eta}_{1}\subset\eta_{4}^{t}\,,\tilde{\eta}_{2}\subset\eta_{5}^{t}\,,\tilde{\eta}_{3}\subset\eta_{6}^{t}\,.

By considering the transpose of the conditions in the second line conditions, we get

η3η~1tη4η~2tη5η~3tη6η~4tη1subscript𝜂3superscriptsubscript~𝜂1𝑡subscript𝜂4superscriptsubscript~𝜂2𝑡subscript𝜂5superscriptsubscript~𝜂3𝑡subscript𝜂6superscriptsubscript~𝜂4𝑡subscript𝜂1absent\displaystyle\eta_{3}\subset\tilde{\eta}_{1}^{t}\subset\eta_{4}\subset\tilde{\eta}_{2}^{t}\subset\eta_{5}\subset\tilde{\eta}_{3}^{t}\subset\eta_{6}\subset\tilde{\eta}_{4}^{t}\subset\eta_{1}\subset
η~5tη2η~6tη3,superscriptsubscript~𝜂5𝑡subscript𝜂2superscriptsubscript~𝜂6𝑡subscript𝜂3\displaystyle\tilde{\eta}_{5}^{t}\subset\eta_{2}\subset\tilde{\eta}_{6}^{t}\subset\eta_{3}\,,

which implies that the summation in (B.7) only receives contributions for

η1=η2=η3=η4=η5=η6=η~1t=η~2t=η~3t=subscript𝜂1subscript𝜂2subscript𝜂3subscript𝜂4subscript𝜂5subscript𝜂6superscriptsubscript~𝜂1𝑡superscriptsubscript~𝜂2𝑡superscriptsubscript~𝜂3𝑡absent\displaystyle\eta_{1}=\eta_{2}=\eta_{3}=\eta_{4}=\eta_{5}=\eta_{6}=\tilde{\eta}_{1}^{t}=\tilde{\eta}_{2}^{t}=\tilde{\eta}_{3}^{t}=
η~4t=η~5t=η~6t,superscriptsubscript~𝜂4𝑡superscriptsubscript~𝜂5𝑡superscriptsubscript~𝜂6𝑡\displaystyle\tilde{\eta}_{4}^{t}=\tilde{\eta}_{5}^{t}=\tilde{\eta}_{6}^{t}\,,

such that (B.7) is reduced to

limnG(Q¯n𝕩,Q¯n𝕪,Q¯n𝕨,Q¯n𝕫)=ηQ¯|η|=k=1(1Q¯k)1.subscript𝑛𝐺superscript¯𝑄𝑛𝕩superscript¯𝑄𝑛𝕪superscript¯𝑄𝑛𝕨superscript¯𝑄𝑛𝕫subscript𝜂superscript¯𝑄𝜂superscriptsubscriptproduct𝑘1superscript1superscript¯𝑄𝑘1\displaystyle\lim_{n\to\infty}G(\bar{Q}^{n}\mathbb{x},\bar{Q}^{n}\mathbb{y},\bar{Q}^{n}\mathbb{w},\bar{Q}^{n}\mathbb{z})=\sum_{\eta}\bar{Q}^{|\eta|}=\prod_{k=1}^{\infty}(1-\bar{Q}^{k})^{-1}\,.

We thus obtain

G(𝕩,𝕪,𝕨,𝕫)=Pk=1(1Q¯k)1,𝐺𝕩𝕪𝕨𝕫subscript𝑃superscriptsubscriptproduct𝑘1superscript1superscript¯𝑄𝑘1G(\mathbb{x},\mathbb{y},\mathbb{w},\mathbb{z})=P_{\infty}\cdot\prod_{k=1}^{\infty}(1-\bar{Q}^{k})^{-1}\,, (B.8)

where the multiplicative prefactor can be worked out to be

P=i=16r,s=1k=1(1QhiQ¯k1xi+3,ryi+5,s)(1QviQ¯k1wi,rzi+3,s)(1QhiQvi1Q¯k1xi+3,rzi+2,s)(1Q~iQ¯k1yi+5,rwi,s)subscript𝑃superscriptsubscriptproduct𝑖16superscriptsubscriptproduct𝑟𝑠1superscriptsubscriptproduct𝑘11subscript𝑄subscript𝑖superscript¯𝑄𝑘1subscript𝑥𝑖3𝑟subscript𝑦𝑖5𝑠1subscript𝑄subscript𝑣𝑖superscript¯𝑄𝑘1subscript𝑤𝑖𝑟subscript𝑧𝑖3𝑠1subscript𝑄subscript𝑖subscript𝑄subscript𝑣𝑖1superscript¯𝑄𝑘1subscript𝑥𝑖3𝑟subscript𝑧𝑖2𝑠1subscript~𝑄𝑖superscript¯𝑄𝑘1subscript𝑦𝑖5𝑟subscript𝑤𝑖𝑠\displaystyle P_{\infty}=\prod_{i=1}^{6}\prod_{r,s=1}^{\infty}\prod_{k=1}^{\infty}\frac{(1-Q_{h_{i}}\bar{Q}^{k-1}x_{i+3,r}y_{i+5,s})(1-Q_{v_{i}}\bar{Q}^{k-1}w_{i,r}z_{i+3,s})}{(1-Q_{h_{i}}Q_{v_{i-1}}\bar{Q}^{k-1}x_{i+3,r}z_{i+2,s})(1-\tilde{Q}_{i}\bar{Q}^{k-1}y_{i+5,r}w_{i,s})}
×(1QhiQ~i1Q¯k1xi+3,ryi+4,s)(1QviQ~i+1Q¯k1wi+1,rzi+3,s)(1QhiQ~i1Qvi2Q¯k1xi+3,rzi+1,s)(1Q~iQ~i+1Q¯k1yi+5,rwi+1,s)absent1subscript𝑄subscript𝑖subscript~𝑄𝑖1superscript¯𝑄𝑘1subscript𝑥𝑖3𝑟subscript𝑦𝑖4𝑠1subscript𝑄subscript𝑣𝑖subscript~𝑄𝑖1superscript¯𝑄𝑘1subscript𝑤𝑖1𝑟subscript𝑧𝑖3𝑠1subscript𝑄subscript𝑖subscript~𝑄𝑖1subscript𝑄subscript𝑣𝑖2superscript¯𝑄𝑘1subscript𝑥𝑖3𝑟subscript𝑧𝑖1𝑠1subscript~𝑄𝑖subscript~𝑄𝑖1superscript¯𝑄𝑘1subscript𝑦𝑖5𝑟subscript𝑤𝑖1𝑠\displaystyle\times\frac{(1-Q_{h_{i}}\tilde{Q}_{i-1}\bar{Q}^{k-1}x_{i+3,r}y_{i+4,s})(1-Q_{v_{i}}\tilde{Q}_{i+1}\bar{Q}^{k-1}w_{i+1,r}z_{i+3,s})}{(1-Q_{h_{i}}\tilde{Q}_{i-1}Q_{v_{i-2}}\bar{Q}^{k-1}x_{i+3,r}z_{i+1,s})(1-\tilde{Q}_{i}\tilde{Q}_{i+1}\bar{Q}^{k-1}y_{i+5,r}w_{i+1,s})}
×(1QhiQ~i1Q~i2Q¯k1xi+3,ryi+3,s)(1QviQ~i+1Q~i+2Q¯k1wi+2,rzi+3,s)(1QhiQ~i1Q~i2Qvi3Q¯k1xi+3,rzi,s)(1Q~iQ~i+1Q~i+2Q¯k1yi+5,rwi+2,s)absent1subscript𝑄subscript𝑖subscript~𝑄𝑖1subscript~𝑄𝑖2superscript¯𝑄𝑘1subscript𝑥𝑖3𝑟subscript𝑦𝑖3𝑠1subscript𝑄subscript𝑣𝑖subscript~𝑄𝑖1subscript~𝑄𝑖2superscript¯𝑄𝑘1subscript𝑤𝑖2𝑟subscript𝑧𝑖3𝑠1subscript𝑄subscript𝑖subscript~𝑄𝑖1subscript~𝑄𝑖2subscript𝑄subscript𝑣𝑖3superscript¯𝑄𝑘1subscript𝑥𝑖3𝑟subscript𝑧𝑖𝑠1subscript~𝑄𝑖subscript~𝑄𝑖1subscript~𝑄𝑖2superscript¯𝑄𝑘1subscript𝑦𝑖5𝑟subscript𝑤𝑖2𝑠\displaystyle\times\frac{(1-Q_{h_{i}}\tilde{Q}_{i-1}\tilde{Q}_{i-2}\bar{Q}^{k-1}x_{i+3,r}y_{i+3,s})(1-Q_{v_{i}}\tilde{Q}_{i+1}\tilde{Q}_{i+2}\bar{Q}^{k-1}w_{i+2,r}z_{i+3,s})}{(1-Q_{h_{i}}\tilde{Q}_{i-1}\tilde{Q}_{i-2}Q_{v_{i-3}}\bar{Q}^{k-1}x_{i+3,r}z_{i,s})(1-\tilde{Q}_{i}\tilde{Q}_{i+1}\tilde{Q}_{i+2}\bar{Q}^{k-1}y_{i+5,r}w_{i+2,s})}
×(1QhiQ~i1Q~i2Q~i3Q¯k1xi+3,ryi+2,s)(1QviQ~i+1Q~i+2Q~i+3Q¯k1wi+3,rzi+3,s)(1QhiQ~i1Q~i2Q~i3Qvi4Q¯k1xi+3,rzi+5,s)(1Q~iQ~i+1Q~i+2Q~i+3Q¯k+5yi+5,rwi+3,s)absent1subscript𝑄subscript𝑖subscript~𝑄𝑖1subscript~𝑄𝑖2subscript~𝑄𝑖3superscript¯𝑄𝑘1subscript𝑥𝑖3𝑟subscript𝑦𝑖2𝑠1subscript𝑄subscript𝑣𝑖subscript~𝑄𝑖1subscript~𝑄𝑖2subscript~𝑄𝑖3superscript¯𝑄𝑘1subscript𝑤𝑖3𝑟subscript𝑧𝑖3𝑠1subscript𝑄subscript𝑖subscript~𝑄𝑖1subscript~𝑄𝑖2subscript~𝑄𝑖3subscript𝑄subscript𝑣𝑖4superscript¯𝑄𝑘1subscript𝑥𝑖3𝑟subscript𝑧𝑖5𝑠1subscript~𝑄𝑖subscript~𝑄𝑖1subscript~𝑄𝑖2subscript~𝑄𝑖3superscript¯𝑄𝑘5subscript𝑦𝑖5𝑟subscript𝑤𝑖3𝑠\displaystyle\times\frac{(1-Q_{h_{i}}\tilde{Q}_{i-1}\tilde{Q}_{i-2}\tilde{Q}_{i-3}\bar{Q}^{k-1}x_{i+3,r}y_{i+2,s})(1-Q_{v_{i}}\tilde{Q}_{i+1}\tilde{Q}_{i+2}\tilde{Q}_{i+3}\bar{Q}^{k-1}w_{i+3,r}z_{i+3,s})}{(1-Q_{h_{i}}\tilde{Q}_{i-1}\tilde{Q}_{i-2}\tilde{Q}_{i-3}Q_{v_{i-4}}\bar{Q}^{k-1}x_{i+3,r}z_{i+5,s})(1-\tilde{Q}_{i}\tilde{Q}_{i+1}\tilde{Q}_{i+2}\tilde{Q}_{i+3}\bar{Q}^{k+5}y_{i+5,r}w_{i+3,s})}
×(1QhiQ~i1Q~i+4Q~i3Q~i+2Q¯k1xi+3,ryi+1,s)(1QviQ~i+1Q~i+2Q~i+3Q~i+4Q¯k1wi+4,rzi+3,s)(1QhiQ~i1Q~i+4Q~i3Q~i4Qvi5Q¯k1xi+3,rzi2,s)(1Q~iQ~i+1Q~i+2Q~i+3Q~i+4Q¯k1yi+5,rwi+4,s)absent1subscript𝑄subscript𝑖subscript~𝑄𝑖1subscript~𝑄𝑖4subscript~𝑄𝑖3subscript~𝑄𝑖2superscript¯𝑄𝑘1subscript𝑥𝑖3𝑟subscript𝑦𝑖1𝑠1subscript𝑄subscript𝑣𝑖subscript~𝑄𝑖1subscript~𝑄𝑖2subscript~𝑄𝑖3subscript~𝑄𝑖4superscript¯𝑄𝑘1subscript𝑤𝑖4𝑟subscript𝑧𝑖3𝑠1subscript𝑄subscript𝑖subscript~𝑄𝑖1subscript~𝑄𝑖4subscript~𝑄𝑖3subscript~𝑄𝑖4subscript𝑄subscript𝑣𝑖5superscript¯𝑄𝑘1subscript𝑥𝑖3𝑟subscript𝑧𝑖2𝑠1subscript~𝑄𝑖subscript~𝑄𝑖1subscript~𝑄𝑖2subscript~𝑄𝑖3subscript~𝑄𝑖4superscript¯𝑄𝑘1subscript𝑦𝑖5𝑟subscript𝑤𝑖4𝑠\displaystyle\times\frac{(1-Q_{h_{i}}\tilde{Q}_{i-1}\tilde{Q}_{i+4}\tilde{Q}_{i-3}\tilde{Q}_{i+2}\bar{Q}^{k-1}x_{i+3,r}y_{i+1,s})(1-Q_{v_{i}}\tilde{Q}_{i+1}\tilde{Q}_{i+2}\tilde{Q}_{i+3}\tilde{Q}_{i+4}\bar{Q}^{k-1}w_{i+4,r}z_{i+3,s})}{(1-Q_{h_{i}}\tilde{Q}_{i-1}\tilde{Q}_{i+4}\tilde{Q}_{i-3}\tilde{Q}_{i-4}Q_{v_{i-5}}\bar{Q}^{k-1}x_{i+3,r}z_{i-2,s})(1-\tilde{Q}_{i}\tilde{Q}_{i+1}\tilde{Q}_{i+2}\tilde{Q}_{i+3}\tilde{Q}_{i+4}\bar{Q}^{k-1}y_{i+5,r}w_{i+4,s})}
×(1QhiQ~i1Q~i2Q~i3Q~i4Q~i5Q¯k1xi+3,ryi,s)(1QviQ~i+1Q~i+2Q~i+3Q~i+4Q~i+5Q¯k1wi+5,rzi+3,s)(1Q¯kxi+3,rzi+3,s)(1Q¯kyi+5,rwi+5,s).absent1subscript𝑄subscript𝑖subscript~𝑄𝑖1subscript~𝑄𝑖2subscript~𝑄𝑖3subscript~𝑄𝑖4subscript~𝑄𝑖5superscript¯𝑄𝑘1subscript𝑥𝑖3𝑟subscript𝑦𝑖𝑠1subscript𝑄subscript𝑣𝑖subscript~𝑄𝑖1subscript~𝑄𝑖2subscript~𝑄𝑖3subscript~𝑄𝑖4subscript~𝑄𝑖5superscript¯𝑄𝑘1subscript𝑤𝑖5𝑟subscript𝑧𝑖3𝑠1superscript¯𝑄𝑘subscript𝑥𝑖3𝑟subscript𝑧𝑖3𝑠1superscript¯𝑄𝑘subscript𝑦𝑖5𝑟subscript𝑤𝑖5𝑠\displaystyle\times\frac{(1-Q_{h_{i}}\tilde{Q}_{i-1}\tilde{Q}_{i-2}\tilde{Q}_{i-3}\tilde{Q}_{i-4}\tilde{Q}_{i-5}\bar{Q}^{k-1}x_{i+3,r}y_{i,s})(1-Q_{v_{i}}\tilde{Q}_{i+1}\tilde{Q}_{i+2}\tilde{Q}_{i+3}\tilde{Q}_{i+4}\tilde{Q}_{i+5}\bar{Q}^{k-1}w_{i+5,r}z_{i+3,s})}{(1-\bar{Q}^{k}x_{i+3,r}z_{i+3,s})(1-\bar{Q}^{k}y_{i+5,r}w_{i+5,s})}\,. (B.9)

Finally, using the consistency conditions (III.3)-(III.8), this expression can be simplified to Eq.(III.18).

Appendix C Flop Transforms for Twisted Diagrams

In this appendix, we recapitulate a series of flop and symmetry transforms of a twisted web diagram of length L𝐿L with generic shift δ𝛿\delta (as shown in Fig. 9 and equivalently in Fig. 10, along with a labelling of all relevant parameters and integer partitions) that relate it to a twisted web diagram of the same length but with shift δ+1𝛿1\delta+1. The duality between the two twisted web diagrams was first discussed in Hohenegger:2016yuv . It can be applied iteratively to obtain a web diagram with shift δ=0𝛿0\delta=0, which was used in Hohenegger:2016yuv to argue for a duality between XN,Msubscript𝑋𝑁𝑀X_{N,M} and XN,Msubscript𝑋superscript𝑁superscript𝑀X_{N^{\prime},M^{\prime}} for NM=NM𝑁𝑀superscript𝑁superscript𝑀NM=N^{\prime}M^{\prime} and gcd(N,M)=k=gcd(N,M)gcd𝑁𝑀𝑘gcdsuperscript𝑁superscript𝑀\text{gcd}(N,M)=k=\text{gcd}(N^{\prime},M^{\prime}). Here, we are primarily interested in the case gcd(N,M)=1gcd𝑁𝑀1\text{gcd}(N,M)=1.

We start out by performing an SL(2,)𝑆𝐿2SL(2,\mathbb{Z}) transformation of the twisted web diagram in Fig. 9, whose result is shown in Fig. 10.

\cdots\cdotsh1subscript1h_{1}h2subscript2h_{2}h3subscript3h_{3}hδsubscript𝛿h_{\delta}hδ+1subscript𝛿1h_{\delta+1}hδ+2subscript𝛿2h_{\delta+2}hL1subscript𝐿1h_{L-1}hLsubscript𝐿h_{L}h1subscript1h_{1}m1subscript𝑚1m_{1}m2subscript𝑚2m_{2}mδsubscript𝑚𝛿m_{\delta}mδ+1subscript𝑚𝛿1m_{\delta+1}mL1subscript𝑚𝐿1m_{L-1}mLsubscript𝑚𝐿m_{L}v1subscript𝑣1v_{1}v2subscript𝑣2v_{2}vδsubscript𝑣𝛿v_{\delta}vδ+1subscript𝑣𝛿1v_{\delta+1}vL1subscript𝑣𝐿1v_{L-1}vLsubscript𝑣𝐿v_{L}a𝑎aa𝑎a111222δ𝛿\deltaδ+1𝛿1\delta+1L1𝐿1L-1L𝐿LLδ+1𝐿𝛿1L-\delta+1Lδ+2𝐿𝛿2L-\delta+2L𝐿L111Lδ1𝐿𝛿1L-\delta-1Lδ𝐿𝛿L-\deltaa^1subscript^𝑎1\widehat{a}_{1}a^2subscript^𝑎2\widehat{a}_{2}\cdotsa^δsubscript^𝑎𝛿\widehat{a}_{\delta}a^δ+1subscript^𝑎𝛿1\widehat{a}_{\delta}+1\cdotsa^L1subscript^𝑎𝐿1\widehat{a}_{L-1}a^Lsubscript^𝑎𝐿\widehat{a}_{L}R(δ)superscript𝑅𝛿-R^{(\delta)}\cdots\cdotsS𝑆S
Figure 10: Twisted web diagram with shift δ𝛿\delta. This diagram is obtained from Fig. 9 through an SL(2,)𝑆𝐿2SL(2,\mathbb{Z}) transformation.

This does not change the Kähler parameters, as indicated in Fig. 10.

As the next step, we perform a flop transformation on the intervals {h2,,hL}subscript2subscript𝐿\{h_{2},\ldots,h_{L}\}. After suitable cutting and re-gluing, the twisted web can be presented in the form shown in Fig. 11. Notice that not only hihisubscript𝑖subscript𝑖h_{i}\longrightarrow-h_{i} for i=2,,L𝑖2𝐿i=2,\ldots,L, but also the remaining parameters have changed according to

m1=m1+hδ+2,subscriptsuperscript𝑚1subscript𝑚1subscript𝛿2\displaystyle m^{\prime}_{1}=m_{1}+h_{\delta+2}\,, v1=v1+h2,subscriptsuperscript𝑣1subscript𝑣1subscript2\displaystyle v^{\prime}_{1}=v_{1}+h_{2}\,,
m2=m2+h2+hδ+3,subscriptsuperscript𝑚2subscript𝑚2subscript2subscript𝛿3\displaystyle m^{\prime}_{2}=m_{2}+h_{2}+h_{\delta+3}\,, v2=v2+h2+h3,subscriptsuperscript𝑣2subscript𝑣2subscript2subscript3\displaystyle v^{\prime}_{2}=v_{2}+h_{2}+h_{3}\,,
\displaystyle\ldots \displaystyle\ldots
mL1=mL1+hL1+hδ,subscriptsuperscript𝑚𝐿1subscript𝑚𝐿1subscript𝐿1subscript𝛿\displaystyle m^{\prime}_{L-1}=m_{L-1}+h_{L-1}+h_{\delta}\,, vL1=vL1+hL1+hL,subscriptsuperscript𝑣𝐿1subscript𝑣𝐿1subscript𝐿1subscript𝐿\displaystyle v^{\prime}_{L-1}=v_{L-1}+h_{L-1}+h_{L}\,,
mL=mL+hL+hδ+1,subscriptsuperscript𝑚𝐿subscript𝑚𝐿subscript𝐿subscript𝛿1\displaystyle m^{\prime}_{L}=m_{L}+h_{L}+h_{\delta+1}\,, vL=vL+hL,subscriptsuperscript𝑣𝐿subscript𝑣𝐿subscript𝐿\displaystyle v^{\prime}_{L}=v_{L}+h_{L}\,, (C.1)

which reflect how the various intervals are connected to the ones that are being flopped.

\cdots\cdotsv1subscriptsuperscript𝑣1v^{\prime}_{1}v2subscriptsuperscript𝑣2v^{\prime}_{2}v3subscriptsuperscript𝑣3v^{\prime}_{3}vδsubscriptsuperscript𝑣𝛿v^{\prime}_{\delta}vδ+1subscriptsuperscript𝑣𝛿1v^{\prime}_{\delta+1}vδ+2subscriptsuperscript𝑣𝛿2v^{\prime}_{\delta+2}vL2subscriptsuperscript𝑣𝐿2v^{\prime}_{L-2}vL1subscriptsuperscript𝑣𝐿1v^{\prime}_{L-1}vLsubscriptsuperscript𝑣𝐿v^{\prime}_{L}m1subscriptsuperscript𝑚1m^{\prime}_{1}m2subscriptsuperscript𝑚2m^{\prime}_{2}m3subscriptsuperscript𝑚3m^{\prime}_{3}mδ+1subscriptsuperscript𝑚𝛿1m^{\prime}_{\delta+1}mδ+2subscriptsuperscript𝑚𝛿2m^{\prime}_{\delta+2}mL1subscriptsuperscript𝑚𝐿1m^{\prime}_{L-1}mLsubscriptsuperscript𝑚𝐿m^{\prime}_{L}h1subscript1h_{1}h2subscript2-h_{2}h3subscript3-h_{3}hδ+1subscript𝛿1-h_{\delta+1}hδ+2subscript𝛿2-h_{\delta+2}hL1subscript𝐿1-h_{L-1}hLsubscript𝐿-h_{L}a𝑎aa𝑎a111222333δ+1𝛿1\delta+1δ+2𝛿2\delta+2L1𝐿1L-1L𝐿LLδ+1𝐿𝛿1L-\delta+1Lδ+2𝐿𝛿2L-\delta+2L𝐿L111Lδ2𝐿𝛿2L-\delta-2Lδ1𝐿𝛿1L-\delta-1Lδ𝐿𝛿L-\delta
Figure 11: Web diagram obtained by flopping h2,,hLsubscript2subscript𝐿h_{2},\ldots,h_{L} in Fig. 10.

As the final step, performing a flop transformation of h1subscript1h_{1}, we get the web shown in Fig. 12, which is a twisted web diagram with shift δ+1𝛿1\delta+1. Concerning the parameters that are associated with the individual line segments in the web, we have

v1′′=v1+h1+h2,superscriptsubscript𝑣1′′subscript𝑣1subscript1subscript2\displaystyle v_{1}^{\prime\prime}=v_{1}+h_{1}+h_{2}\,, vL′′=vL+h1+hL,subscriptsuperscript𝑣′′𝐿subscript𝑣𝐿subscript1subscript𝐿\displaystyle v^{\prime\prime}_{L}=v_{L}+h_{1}+h_{L}\,, m1′′=m1+h1+hδ+2,subscriptsuperscript𝑚′′1subscript𝑚1subscript1subscript𝛿2\displaystyle m^{\prime\prime}_{1}=m_{1}+h_{1}+h_{\delta+2}\,, mLδ′′=mLδ+h1+hLδ,subscriptsuperscript𝑚′′𝐿𝛿subscript𝑚𝐿𝛿subscript1subscript𝐿𝛿\displaystyle m^{\prime\prime}_{L-\delta}=m_{L-\delta}+h_{1}+h_{L-\delta}\,, (C.2)

and

vi′′=vi,subscriptsuperscript𝑣′′𝑖subscriptsuperscript𝑣𝑖\displaystyle v^{\prime\prime}_{i}=v^{\prime}_{i}\,, hi′′=hi,subscriptsuperscript′′𝑖subscriptsuperscript𝑖\displaystyle h^{\prime\prime}_{i}=h^{\prime}_{i}\,, mi′′=mi,subscriptsuperscript𝑚′′𝑖subscriptsuperscript𝑚𝑖\displaystyle m^{\prime\prime}_{i}=m^{\prime}_{i}\,, i{1,L}.for-all𝑖1𝐿\displaystyle\forall i\notin\{1,L\}\,. (C.3)
\cdots\cdotsvL′′subscriptsuperscript𝑣′′𝐿v^{\prime\prime}_{L}v1′′subscriptsuperscript𝑣′′1v^{\prime\prime}_{1}v2′′subscriptsuperscript𝑣′′2v^{\prime\prime}_{2}vδ′′subscriptsuperscript𝑣′′𝛿v^{\prime\prime}_{\delta}vδ+1′′subscriptsuperscript𝑣′′𝛿1v^{\prime\prime}_{\delta+1}vδ+2′′subscriptsuperscript𝑣′′𝛿2v^{\prime\prime}_{\delta+2}vL2′′subscriptsuperscript𝑣′′𝐿2v^{\prime\prime}_{L-2}vL1′′subscriptsuperscript𝑣′′𝐿1v^{\prime\prime}_{L-1}vL′′subscriptsuperscript𝑣′′𝐿v^{\prime\prime}_{L}m1′′subscriptsuperscript𝑚′′1m^{\prime\prime}_{1}m2′′subscriptsuperscript𝑚′′2m^{\prime\prime}_{2}mδ+1′′subscriptsuperscript𝑚′′𝛿1m^{\prime\prime}_{\delta+1}mδ+2′′subscriptsuperscript𝑚′′𝛿2m^{\prime\prime}_{\delta+2}mL1′′subscriptsuperscript𝑚′′𝐿1m^{\prime\prime}_{L-1}mL′′subscriptsuperscript𝑚′′𝐿m^{\prime\prime}_{L}h1subscript1-h_{1}h2subscript2-h_{2}hδ+1subscript𝛿1-h_{\delta+1}hδ+2subscript𝛿2-h_{\delta+2}hL1subscript𝐿1-h_{L-1}hLsubscript𝐿-h_{L}a𝑎aa𝑎a111222δ+1𝛿1\delta+1δ+2𝛿2\delta+2L1𝐿1L-1L𝐿LLδ𝐿𝛿L-\deltaLδ+1𝐿𝛿1L-\delta+1L𝐿L111Lδ2𝐿𝛿2L-\delta-2Lδ1𝐿𝛿1L-\delta-1a^1subscript^𝑎1\widehat{a}_{1}a^2subscript^𝑎2\widehat{a}_{2}\cdotsa^δsubscript^𝑎𝛿\widehat{a}_{\delta}a^δ+1subscript^𝑎𝛿1\widehat{a}_{\delta}+1\cdotsa^L1subscript^𝑎𝐿1\widehat{a}_{L-1}a^Lsubscript^𝑎𝐿\widehat{a}_{L}R(δ+1)superscript𝑅𝛿1-R^{(\delta+1)}\cdots\cdotsS𝑆S
Figure 12: Twisted web diagram with shift δ+1𝛿1\delta+1 obtained from Fig. 11 through flop of the curve h1subscript1h_{1}.

We can summarise this in the form of the duality map

vihi,subscript𝑣𝑖subscript𝑖\displaystyle v_{i}\longrightarrow-h_{i}\,,
hivi1+hi+hi1,subscript𝑖subscript𝑣𝑖1subscript𝑖subscript𝑖1\displaystyle h_{i}\longrightarrow v_{i-1}+h_{i}+h_{i-1}\,,
mimi+hi+hi+δ+1.subscript𝑚𝑖subscript𝑚𝑖subscript𝑖subscript𝑖𝛿1\displaystyle m_{i}\longrightarrow m_{i}+h_{i}+h_{i+\delta+1}\,. (C.4)

Moreover, the basis parameters a^1,,Lsubscript^𝑎1𝐿\widehat{a}_{1,\ldots,L} and S𝑆S remain the same as in Fig. 10 (i.e. they are invariant under the duality map), while the parameters R(δ)superscript𝑅𝛿R^{(\delta)} and R(δ+1)superscript𝑅𝛿1R^{(\delta+1)} are related by

R(δ+1)R(δ)=S.superscript𝑅𝛿1superscript𝑅𝛿𝑆\displaystyle R^{(\delta+1)}-R^{(\delta)}=S\,. (C.5)

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